Why is this definition of prime natural?












3














Definition: An element $p$, not zero and not a unit, is called prime if $p|ab$ implies $p|a$ or $p|b$.



I am having trouble understanding why this definition of prime follows naturally. Could someone provide an intuitive explanation for this definition? For context, the text I am learning from has briefly gone over groups, rings, and integral domains.



Thank you!










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Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • 1




    You may find this article on Euclid's lemma useful.
    – André 3000
    3 hours ago






  • 1




    Well, intuitive, prime to me means indivisible. If $p|ab$ but it's impossible to split $p$ so that part of it goes into $a$ and part of it goes into $b$ then... it's indivisible.
    – fleablood
    3 hours ago






  • 1




    I enjoyed reading this which gives some intuition on what we aiming to do with primes.
    – Mason
    3 hours ago






  • 1




    This property is precisely what is required of irreducibles in order for factorizations into irreducibles to be unique (in integral domains).
    – Bill Dubuque
    3 hours ago


















3














Definition: An element $p$, not zero and not a unit, is called prime if $p|ab$ implies $p|a$ or $p|b$.



I am having trouble understanding why this definition of prime follows naturally. Could someone provide an intuitive explanation for this definition? For context, the text I am learning from has briefly gone over groups, rings, and integral domains.



Thank you!










share|cite|improve this question







New contributor




Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
















  • 1




    You may find this article on Euclid's lemma useful.
    – André 3000
    3 hours ago






  • 1




    Well, intuitive, prime to me means indivisible. If $p|ab$ but it's impossible to split $p$ so that part of it goes into $a$ and part of it goes into $b$ then... it's indivisible.
    – fleablood
    3 hours ago






  • 1




    I enjoyed reading this which gives some intuition on what we aiming to do with primes.
    – Mason
    3 hours ago






  • 1




    This property is precisely what is required of irreducibles in order for factorizations into irreducibles to be unique (in integral domains).
    – Bill Dubuque
    3 hours ago
















3












3








3


1





Definition: An element $p$, not zero and not a unit, is called prime if $p|ab$ implies $p|a$ or $p|b$.



I am having trouble understanding why this definition of prime follows naturally. Could someone provide an intuitive explanation for this definition? For context, the text I am learning from has briefly gone over groups, rings, and integral domains.



Thank you!










share|cite|improve this question







New contributor




Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











Definition: An element $p$, not zero and not a unit, is called prime if $p|ab$ implies $p|a$ or $p|b$.



I am having trouble understanding why this definition of prime follows naturally. Could someone provide an intuitive explanation for this definition? For context, the text I am learning from has briefly gone over groups, rings, and integral domains.



Thank you!







elementary-number-theory ring-theory prime-numbers integral-domain






share|cite|improve this question







New contributor




Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question







New contributor




Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question






New contributor




Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 4 hours ago









WilsonWilson

182




182




New contributor




Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Wilson is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    You may find this article on Euclid's lemma useful.
    – André 3000
    3 hours ago






  • 1




    Well, intuitive, prime to me means indivisible. If $p|ab$ but it's impossible to split $p$ so that part of it goes into $a$ and part of it goes into $b$ then... it's indivisible.
    – fleablood
    3 hours ago






  • 1




    I enjoyed reading this which gives some intuition on what we aiming to do with primes.
    – Mason
    3 hours ago






  • 1




    This property is precisely what is required of irreducibles in order for factorizations into irreducibles to be unique (in integral domains).
    – Bill Dubuque
    3 hours ago
















  • 1




    You may find this article on Euclid's lemma useful.
    – André 3000
    3 hours ago






  • 1




    Well, intuitive, prime to me means indivisible. If $p|ab$ but it's impossible to split $p$ so that part of it goes into $a$ and part of it goes into $b$ then... it's indivisible.
    – fleablood
    3 hours ago






  • 1




    I enjoyed reading this which gives some intuition on what we aiming to do with primes.
    – Mason
    3 hours ago






  • 1




    This property is precisely what is required of irreducibles in order for factorizations into irreducibles to be unique (in integral domains).
    – Bill Dubuque
    3 hours ago










1




1




You may find this article on Euclid's lemma useful.
– André 3000
3 hours ago




You may find this article on Euclid's lemma useful.
– André 3000
3 hours ago




1




1




Well, intuitive, prime to me means indivisible. If $p|ab$ but it's impossible to split $p$ so that part of it goes into $a$ and part of it goes into $b$ then... it's indivisible.
– fleablood
3 hours ago




Well, intuitive, prime to me means indivisible. If $p|ab$ but it's impossible to split $p$ so that part of it goes into $a$ and part of it goes into $b$ then... it's indivisible.
– fleablood
3 hours ago




1




1




I enjoyed reading this which gives some intuition on what we aiming to do with primes.
– Mason
3 hours ago




I enjoyed reading this which gives some intuition on what we aiming to do with primes.
– Mason
3 hours ago




1




1




This property is precisely what is required of irreducibles in order for factorizations into irreducibles to be unique (in integral domains).
– Bill Dubuque
3 hours ago






This property is precisely what is required of irreducibles in order for factorizations into irreducibles to be unique (in integral domains).
– Bill Dubuque
3 hours ago












3 Answers
3






active

oldest

votes


















4














Good question!



There are probably several possible answers to this question, but here is my perspective.



Reason 1: Ring Theoretic



Let $R$ be a commutative ring. Let $fin R$, then consider $fR={af : ain R}$.
You can check that this is an ideal of $R$, and it is also often denoted by simply $(f)$ (meaning the ideal of $R$ generated by $f$).



Now we can ask the question




When is $R/fR$ a domain?




It turns out the answer is:




Precisely when $f$ is prime.




Proof:



If $p$ is prime, then if $$abequiv 0 pmod{pR},$$
by definition this means $pmid ab$, which since $p$ is prime implies that $pmid a$ or $pmid b$. However this in turn means
$$aequiv 0!!pmod{pR}quadtext{or}quad bequiv 0!!pmod{pR},$$
which is what it means for $R/pR$ to be a domain.



Conversely, if $R/pR$ is a domain, then if $pmid ab$, $abequiv 0 pmod{pR}$, so either $aequiv 0 pmod{pR}$ or $bequiv 0 pmod{pR}$, which means either $pmid a$ or $pmid b$. Hence $p$ is prime. $blacksquare$



Reason 2: Number Theoretic (kind of)



The other way we come up with this naturally is that it is the condition we need to hold in order to get unique factorizations.



I.e., suppose we have two factorizations of an element $xin R$ into irreducibles
$$x = prod_i p_i = prod_j q_j,$$
with $p_i,q_jin R$ irreducibles,
then when are we guaranteed that some $p_i$ occurring in the first factorization appears somewhere on the left hand side (or an associate of $p_i$, i.e. a unit times $p_i$, since for example in the integers we could have $9=3cdot 3 = (-3)cdot (-3)$)?



Well, we need to have $p_i$ divide one of the $q_j$ (for then they are associates, since $p_i$ and $q_j$ are irreducibles).



The condition that for all multiples $x$ of $p_i$, $p_i$ divides some $q_j$ for any factorization $x=prod_j q_j$ of $x$ into irreducibles is equivalent to $p_i$ being prime (for a Noetherian ring, so that we are guaranteed to have factorizations into irreducibles, otherwise bad things could happen).



Proof:



A note on notation: I'll replace $p_i$ with $p$.



Suppose $p$ is prime, and $pmid x$, and $prod_j q_j$ is a factorization of $x$ into irreducibles, then we induct on the length of the factorization. If $x=q_1$ is irreducible, then $pmid q_1$ by definition, and we are done. Otherwise, since $pmid (q_1cdots q_{n-1})q_n$, then by primality of $p$, either $pmid q_1cdots q_{n-1}$, in which case $pmid q_j$ for some $j$ by the inductive hypothesis, or $pmid q_n$, and we are done.



Conversely, if $p$ has the property discussed above, then if $pmid ab$ for some $a$ and $b$, then let $a=prod_ialpha_i$ and $b=prod_jbeta_j$ be factorizations of $a$ and $b$ into irreducibles (since $R$ is Noetherian. If you aren't familiar with Noetherianness yet, then just take the existence of factorizations into irreducibles as a black box for now). Then $pmid x ab= prod_i alpha_i prod_j beta_j$, so by the property we're assuming $p$ has, either $pmid alpha_i$ for some $i$, or $pmid beta_j$ for some $j$, and thus either $pmid a$ or $pmid b$. Hence $p$ is prime.



Reason 3: (Actually a consequence of reason 2)



There is a theorem, which is relevant here.




A Noetherian domain $R$ is a UFD if and only if every irreducible is prime.







share|cite|improve this answer























  • Could you clarify your choice of representation of factorization? Why is there a factor of $u$ and $v$ separate from the product of the rest of the irreducible factors? Also, could you expand upon this "That $p_{i}$ divides some $q_{j}$ for all factorizations of multiples of $p_{i}$ is equivalent to $p_{i}$ being prime?" I am a little lost. Thank you!
    – Wilson
    3 hours ago










  • @Wilson, sorry yes. I'll edit in a moment, but $u$ and $v$ are units, though I could have omitted them.
    – jgon
    3 hours ago










  • Also, what does the notation $R/(p)$ and $R/(f)$ denote in this context? Sorry, I am a little new to these topics. Thank you again.
    – Wilson
    3 hours ago










  • @Wilson No worries. I'm back to edit now
    – jgon
    3 hours ago










  • @Wilson I've edited. Hopefully it will be more clear now :)
    – jgon
    3 hours ago



















1














An ideal is prime if and only if its set-theoretical complement is multiplicatively closed. (We want to keep obtaining non-multiples of $p$ when multiplying non-multiples of $p$.)



This allows for constructing a new ring (addition and multiplication) of “fractions” whose “denominators” live only in the complement of the prime ideal (such construction is called “localization”). This new ring is simpler than the original ring, exposing only “local behaviour”.






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    0














    Welcome!
    take 6 for example it is obviously not prime. We have that $6|6=2*3$ but $6nmid 2$ and $6nmid 3$.



    So if this definition is not true, that means there must be "a part" of the number hidden as a factor of $a$ and a different part hidden in $b$. Which whould of course mean that the number itshelf has more than one factor so it is not prime in the classical sence






    share|cite|improve this answer





















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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      4














      Good question!



      There are probably several possible answers to this question, but here is my perspective.



      Reason 1: Ring Theoretic



      Let $R$ be a commutative ring. Let $fin R$, then consider $fR={af : ain R}$.
      You can check that this is an ideal of $R$, and it is also often denoted by simply $(f)$ (meaning the ideal of $R$ generated by $f$).



      Now we can ask the question




      When is $R/fR$ a domain?




      It turns out the answer is:




      Precisely when $f$ is prime.




      Proof:



      If $p$ is prime, then if $$abequiv 0 pmod{pR},$$
      by definition this means $pmid ab$, which since $p$ is prime implies that $pmid a$ or $pmid b$. However this in turn means
      $$aequiv 0!!pmod{pR}quadtext{or}quad bequiv 0!!pmod{pR},$$
      which is what it means for $R/pR$ to be a domain.



      Conversely, if $R/pR$ is a domain, then if $pmid ab$, $abequiv 0 pmod{pR}$, so either $aequiv 0 pmod{pR}$ or $bequiv 0 pmod{pR}$, which means either $pmid a$ or $pmid b$. Hence $p$ is prime. $blacksquare$



      Reason 2: Number Theoretic (kind of)



      The other way we come up with this naturally is that it is the condition we need to hold in order to get unique factorizations.



      I.e., suppose we have two factorizations of an element $xin R$ into irreducibles
      $$x = prod_i p_i = prod_j q_j,$$
      with $p_i,q_jin R$ irreducibles,
      then when are we guaranteed that some $p_i$ occurring in the first factorization appears somewhere on the left hand side (or an associate of $p_i$, i.e. a unit times $p_i$, since for example in the integers we could have $9=3cdot 3 = (-3)cdot (-3)$)?



      Well, we need to have $p_i$ divide one of the $q_j$ (for then they are associates, since $p_i$ and $q_j$ are irreducibles).



      The condition that for all multiples $x$ of $p_i$, $p_i$ divides some $q_j$ for any factorization $x=prod_j q_j$ of $x$ into irreducibles is equivalent to $p_i$ being prime (for a Noetherian ring, so that we are guaranteed to have factorizations into irreducibles, otherwise bad things could happen).



      Proof:



      A note on notation: I'll replace $p_i$ with $p$.



      Suppose $p$ is prime, and $pmid x$, and $prod_j q_j$ is a factorization of $x$ into irreducibles, then we induct on the length of the factorization. If $x=q_1$ is irreducible, then $pmid q_1$ by definition, and we are done. Otherwise, since $pmid (q_1cdots q_{n-1})q_n$, then by primality of $p$, either $pmid q_1cdots q_{n-1}$, in which case $pmid q_j$ for some $j$ by the inductive hypothesis, or $pmid q_n$, and we are done.



      Conversely, if $p$ has the property discussed above, then if $pmid ab$ for some $a$ and $b$, then let $a=prod_ialpha_i$ and $b=prod_jbeta_j$ be factorizations of $a$ and $b$ into irreducibles (since $R$ is Noetherian. If you aren't familiar with Noetherianness yet, then just take the existence of factorizations into irreducibles as a black box for now). Then $pmid x ab= prod_i alpha_i prod_j beta_j$, so by the property we're assuming $p$ has, either $pmid alpha_i$ for some $i$, or $pmid beta_j$ for some $j$, and thus either $pmid a$ or $pmid b$. Hence $p$ is prime.



      Reason 3: (Actually a consequence of reason 2)



      There is a theorem, which is relevant here.




      A Noetherian domain $R$ is a UFD if and only if every irreducible is prime.







      share|cite|improve this answer























      • Could you clarify your choice of representation of factorization? Why is there a factor of $u$ and $v$ separate from the product of the rest of the irreducible factors? Also, could you expand upon this "That $p_{i}$ divides some $q_{j}$ for all factorizations of multiples of $p_{i}$ is equivalent to $p_{i}$ being prime?" I am a little lost. Thank you!
        – Wilson
        3 hours ago










      • @Wilson, sorry yes. I'll edit in a moment, but $u$ and $v$ are units, though I could have omitted them.
        – jgon
        3 hours ago










      • Also, what does the notation $R/(p)$ and $R/(f)$ denote in this context? Sorry, I am a little new to these topics. Thank you again.
        – Wilson
        3 hours ago










      • @Wilson No worries. I'm back to edit now
        – jgon
        3 hours ago










      • @Wilson I've edited. Hopefully it will be more clear now :)
        – jgon
        3 hours ago
















      4














      Good question!



      There are probably several possible answers to this question, but here is my perspective.



      Reason 1: Ring Theoretic



      Let $R$ be a commutative ring. Let $fin R$, then consider $fR={af : ain R}$.
      You can check that this is an ideal of $R$, and it is also often denoted by simply $(f)$ (meaning the ideal of $R$ generated by $f$).



      Now we can ask the question




      When is $R/fR$ a domain?




      It turns out the answer is:




      Precisely when $f$ is prime.




      Proof:



      If $p$ is prime, then if $$abequiv 0 pmod{pR},$$
      by definition this means $pmid ab$, which since $p$ is prime implies that $pmid a$ or $pmid b$. However this in turn means
      $$aequiv 0!!pmod{pR}quadtext{or}quad bequiv 0!!pmod{pR},$$
      which is what it means for $R/pR$ to be a domain.



      Conversely, if $R/pR$ is a domain, then if $pmid ab$, $abequiv 0 pmod{pR}$, so either $aequiv 0 pmod{pR}$ or $bequiv 0 pmod{pR}$, which means either $pmid a$ or $pmid b$. Hence $p$ is prime. $blacksquare$



      Reason 2: Number Theoretic (kind of)



      The other way we come up with this naturally is that it is the condition we need to hold in order to get unique factorizations.



      I.e., suppose we have two factorizations of an element $xin R$ into irreducibles
      $$x = prod_i p_i = prod_j q_j,$$
      with $p_i,q_jin R$ irreducibles,
      then when are we guaranteed that some $p_i$ occurring in the first factorization appears somewhere on the left hand side (or an associate of $p_i$, i.e. a unit times $p_i$, since for example in the integers we could have $9=3cdot 3 = (-3)cdot (-3)$)?



      Well, we need to have $p_i$ divide one of the $q_j$ (for then they are associates, since $p_i$ and $q_j$ are irreducibles).



      The condition that for all multiples $x$ of $p_i$, $p_i$ divides some $q_j$ for any factorization $x=prod_j q_j$ of $x$ into irreducibles is equivalent to $p_i$ being prime (for a Noetherian ring, so that we are guaranteed to have factorizations into irreducibles, otherwise bad things could happen).



      Proof:



      A note on notation: I'll replace $p_i$ with $p$.



      Suppose $p$ is prime, and $pmid x$, and $prod_j q_j$ is a factorization of $x$ into irreducibles, then we induct on the length of the factorization. If $x=q_1$ is irreducible, then $pmid q_1$ by definition, and we are done. Otherwise, since $pmid (q_1cdots q_{n-1})q_n$, then by primality of $p$, either $pmid q_1cdots q_{n-1}$, in which case $pmid q_j$ for some $j$ by the inductive hypothesis, or $pmid q_n$, and we are done.



      Conversely, if $p$ has the property discussed above, then if $pmid ab$ for some $a$ and $b$, then let $a=prod_ialpha_i$ and $b=prod_jbeta_j$ be factorizations of $a$ and $b$ into irreducibles (since $R$ is Noetherian. If you aren't familiar with Noetherianness yet, then just take the existence of factorizations into irreducibles as a black box for now). Then $pmid x ab= prod_i alpha_i prod_j beta_j$, so by the property we're assuming $p$ has, either $pmid alpha_i$ for some $i$, or $pmid beta_j$ for some $j$, and thus either $pmid a$ or $pmid b$. Hence $p$ is prime.



      Reason 3: (Actually a consequence of reason 2)



      There is a theorem, which is relevant here.




      A Noetherian domain $R$ is a UFD if and only if every irreducible is prime.







      share|cite|improve this answer























      • Could you clarify your choice of representation of factorization? Why is there a factor of $u$ and $v$ separate from the product of the rest of the irreducible factors? Also, could you expand upon this "That $p_{i}$ divides some $q_{j}$ for all factorizations of multiples of $p_{i}$ is equivalent to $p_{i}$ being prime?" I am a little lost. Thank you!
        – Wilson
        3 hours ago










      • @Wilson, sorry yes. I'll edit in a moment, but $u$ and $v$ are units, though I could have omitted them.
        – jgon
        3 hours ago










      • Also, what does the notation $R/(p)$ and $R/(f)$ denote in this context? Sorry, I am a little new to these topics. Thank you again.
        – Wilson
        3 hours ago










      • @Wilson No worries. I'm back to edit now
        – jgon
        3 hours ago










      • @Wilson I've edited. Hopefully it will be more clear now :)
        – jgon
        3 hours ago














      4












      4








      4






      Good question!



      There are probably several possible answers to this question, but here is my perspective.



      Reason 1: Ring Theoretic



      Let $R$ be a commutative ring. Let $fin R$, then consider $fR={af : ain R}$.
      You can check that this is an ideal of $R$, and it is also often denoted by simply $(f)$ (meaning the ideal of $R$ generated by $f$).



      Now we can ask the question




      When is $R/fR$ a domain?




      It turns out the answer is:




      Precisely when $f$ is prime.




      Proof:



      If $p$ is prime, then if $$abequiv 0 pmod{pR},$$
      by definition this means $pmid ab$, which since $p$ is prime implies that $pmid a$ or $pmid b$. However this in turn means
      $$aequiv 0!!pmod{pR}quadtext{or}quad bequiv 0!!pmod{pR},$$
      which is what it means for $R/pR$ to be a domain.



      Conversely, if $R/pR$ is a domain, then if $pmid ab$, $abequiv 0 pmod{pR}$, so either $aequiv 0 pmod{pR}$ or $bequiv 0 pmod{pR}$, which means either $pmid a$ or $pmid b$. Hence $p$ is prime. $blacksquare$



      Reason 2: Number Theoretic (kind of)



      The other way we come up with this naturally is that it is the condition we need to hold in order to get unique factorizations.



      I.e., suppose we have two factorizations of an element $xin R$ into irreducibles
      $$x = prod_i p_i = prod_j q_j,$$
      with $p_i,q_jin R$ irreducibles,
      then when are we guaranteed that some $p_i$ occurring in the first factorization appears somewhere on the left hand side (or an associate of $p_i$, i.e. a unit times $p_i$, since for example in the integers we could have $9=3cdot 3 = (-3)cdot (-3)$)?



      Well, we need to have $p_i$ divide one of the $q_j$ (for then they are associates, since $p_i$ and $q_j$ are irreducibles).



      The condition that for all multiples $x$ of $p_i$, $p_i$ divides some $q_j$ for any factorization $x=prod_j q_j$ of $x$ into irreducibles is equivalent to $p_i$ being prime (for a Noetherian ring, so that we are guaranteed to have factorizations into irreducibles, otherwise bad things could happen).



      Proof:



      A note on notation: I'll replace $p_i$ with $p$.



      Suppose $p$ is prime, and $pmid x$, and $prod_j q_j$ is a factorization of $x$ into irreducibles, then we induct on the length of the factorization. If $x=q_1$ is irreducible, then $pmid q_1$ by definition, and we are done. Otherwise, since $pmid (q_1cdots q_{n-1})q_n$, then by primality of $p$, either $pmid q_1cdots q_{n-1}$, in which case $pmid q_j$ for some $j$ by the inductive hypothesis, or $pmid q_n$, and we are done.



      Conversely, if $p$ has the property discussed above, then if $pmid ab$ for some $a$ and $b$, then let $a=prod_ialpha_i$ and $b=prod_jbeta_j$ be factorizations of $a$ and $b$ into irreducibles (since $R$ is Noetherian. If you aren't familiar with Noetherianness yet, then just take the existence of factorizations into irreducibles as a black box for now). Then $pmid x ab= prod_i alpha_i prod_j beta_j$, so by the property we're assuming $p$ has, either $pmid alpha_i$ for some $i$, or $pmid beta_j$ for some $j$, and thus either $pmid a$ or $pmid b$. Hence $p$ is prime.



      Reason 3: (Actually a consequence of reason 2)



      There is a theorem, which is relevant here.




      A Noetherian domain $R$ is a UFD if and only if every irreducible is prime.







      share|cite|improve this answer














      Good question!



      There are probably several possible answers to this question, but here is my perspective.



      Reason 1: Ring Theoretic



      Let $R$ be a commutative ring. Let $fin R$, then consider $fR={af : ain R}$.
      You can check that this is an ideal of $R$, and it is also often denoted by simply $(f)$ (meaning the ideal of $R$ generated by $f$).



      Now we can ask the question




      When is $R/fR$ a domain?




      It turns out the answer is:




      Precisely when $f$ is prime.




      Proof:



      If $p$ is prime, then if $$abequiv 0 pmod{pR},$$
      by definition this means $pmid ab$, which since $p$ is prime implies that $pmid a$ or $pmid b$. However this in turn means
      $$aequiv 0!!pmod{pR}quadtext{or}quad bequiv 0!!pmod{pR},$$
      which is what it means for $R/pR$ to be a domain.



      Conversely, if $R/pR$ is a domain, then if $pmid ab$, $abequiv 0 pmod{pR}$, so either $aequiv 0 pmod{pR}$ or $bequiv 0 pmod{pR}$, which means either $pmid a$ or $pmid b$. Hence $p$ is prime. $blacksquare$



      Reason 2: Number Theoretic (kind of)



      The other way we come up with this naturally is that it is the condition we need to hold in order to get unique factorizations.



      I.e., suppose we have two factorizations of an element $xin R$ into irreducibles
      $$x = prod_i p_i = prod_j q_j,$$
      with $p_i,q_jin R$ irreducibles,
      then when are we guaranteed that some $p_i$ occurring in the first factorization appears somewhere on the left hand side (or an associate of $p_i$, i.e. a unit times $p_i$, since for example in the integers we could have $9=3cdot 3 = (-3)cdot (-3)$)?



      Well, we need to have $p_i$ divide one of the $q_j$ (for then they are associates, since $p_i$ and $q_j$ are irreducibles).



      The condition that for all multiples $x$ of $p_i$, $p_i$ divides some $q_j$ for any factorization $x=prod_j q_j$ of $x$ into irreducibles is equivalent to $p_i$ being prime (for a Noetherian ring, so that we are guaranteed to have factorizations into irreducibles, otherwise bad things could happen).



      Proof:



      A note on notation: I'll replace $p_i$ with $p$.



      Suppose $p$ is prime, and $pmid x$, and $prod_j q_j$ is a factorization of $x$ into irreducibles, then we induct on the length of the factorization. If $x=q_1$ is irreducible, then $pmid q_1$ by definition, and we are done. Otherwise, since $pmid (q_1cdots q_{n-1})q_n$, then by primality of $p$, either $pmid q_1cdots q_{n-1}$, in which case $pmid q_j$ for some $j$ by the inductive hypothesis, or $pmid q_n$, and we are done.



      Conversely, if $p$ has the property discussed above, then if $pmid ab$ for some $a$ and $b$, then let $a=prod_ialpha_i$ and $b=prod_jbeta_j$ be factorizations of $a$ and $b$ into irreducibles (since $R$ is Noetherian. If you aren't familiar with Noetherianness yet, then just take the existence of factorizations into irreducibles as a black box for now). Then $pmid x ab= prod_i alpha_i prod_j beta_j$, so by the property we're assuming $p$ has, either $pmid alpha_i$ for some $i$, or $pmid beta_j$ for some $j$, and thus either $pmid a$ or $pmid b$. Hence $p$ is prime.



      Reason 3: (Actually a consequence of reason 2)



      There is a theorem, which is relevant here.




      A Noetherian domain $R$ is a UFD if and only if every irreducible is prime.








      share|cite|improve this answer














      share|cite|improve this answer



      share|cite|improve this answer








      edited 3 hours ago

























      answered 3 hours ago









      jgonjgon

      13.3k21941




      13.3k21941












      • Could you clarify your choice of representation of factorization? Why is there a factor of $u$ and $v$ separate from the product of the rest of the irreducible factors? Also, could you expand upon this "That $p_{i}$ divides some $q_{j}$ for all factorizations of multiples of $p_{i}$ is equivalent to $p_{i}$ being prime?" I am a little lost. Thank you!
        – Wilson
        3 hours ago










      • @Wilson, sorry yes. I'll edit in a moment, but $u$ and $v$ are units, though I could have omitted them.
        – jgon
        3 hours ago










      • Also, what does the notation $R/(p)$ and $R/(f)$ denote in this context? Sorry, I am a little new to these topics. Thank you again.
        – Wilson
        3 hours ago










      • @Wilson No worries. I'm back to edit now
        – jgon
        3 hours ago










      • @Wilson I've edited. Hopefully it will be more clear now :)
        – jgon
        3 hours ago


















      • Could you clarify your choice of representation of factorization? Why is there a factor of $u$ and $v$ separate from the product of the rest of the irreducible factors? Also, could you expand upon this "That $p_{i}$ divides some $q_{j}$ for all factorizations of multiples of $p_{i}$ is equivalent to $p_{i}$ being prime?" I am a little lost. Thank you!
        – Wilson
        3 hours ago










      • @Wilson, sorry yes. I'll edit in a moment, but $u$ and $v$ are units, though I could have omitted them.
        – jgon
        3 hours ago










      • Also, what does the notation $R/(p)$ and $R/(f)$ denote in this context? Sorry, I am a little new to these topics. Thank you again.
        – Wilson
        3 hours ago










      • @Wilson No worries. I'm back to edit now
        – jgon
        3 hours ago










      • @Wilson I've edited. Hopefully it will be more clear now :)
        – jgon
        3 hours ago
















      Could you clarify your choice of representation of factorization? Why is there a factor of $u$ and $v$ separate from the product of the rest of the irreducible factors? Also, could you expand upon this "That $p_{i}$ divides some $q_{j}$ for all factorizations of multiples of $p_{i}$ is equivalent to $p_{i}$ being prime?" I am a little lost. Thank you!
      – Wilson
      3 hours ago




      Could you clarify your choice of representation of factorization? Why is there a factor of $u$ and $v$ separate from the product of the rest of the irreducible factors? Also, could you expand upon this "That $p_{i}$ divides some $q_{j}$ for all factorizations of multiples of $p_{i}$ is equivalent to $p_{i}$ being prime?" I am a little lost. Thank you!
      – Wilson
      3 hours ago












      @Wilson, sorry yes. I'll edit in a moment, but $u$ and $v$ are units, though I could have omitted them.
      – jgon
      3 hours ago




      @Wilson, sorry yes. I'll edit in a moment, but $u$ and $v$ are units, though I could have omitted them.
      – jgon
      3 hours ago












      Also, what does the notation $R/(p)$ and $R/(f)$ denote in this context? Sorry, I am a little new to these topics. Thank you again.
      – Wilson
      3 hours ago




      Also, what does the notation $R/(p)$ and $R/(f)$ denote in this context? Sorry, I am a little new to these topics. Thank you again.
      – Wilson
      3 hours ago












      @Wilson No worries. I'm back to edit now
      – jgon
      3 hours ago




      @Wilson No worries. I'm back to edit now
      – jgon
      3 hours ago












      @Wilson I've edited. Hopefully it will be more clear now :)
      – jgon
      3 hours ago




      @Wilson I've edited. Hopefully it will be more clear now :)
      – jgon
      3 hours ago











      1














      An ideal is prime if and only if its set-theoretical complement is multiplicatively closed. (We want to keep obtaining non-multiples of $p$ when multiplying non-multiples of $p$.)



      This allows for constructing a new ring (addition and multiplication) of “fractions” whose “denominators” live only in the complement of the prime ideal (such construction is called “localization”). This new ring is simpler than the original ring, exposing only “local behaviour”.






      share|cite|improve this answer


























        1














        An ideal is prime if and only if its set-theoretical complement is multiplicatively closed. (We want to keep obtaining non-multiples of $p$ when multiplying non-multiples of $p$.)



        This allows for constructing a new ring (addition and multiplication) of “fractions” whose “denominators” live only in the complement of the prime ideal (such construction is called “localization”). This new ring is simpler than the original ring, exposing only “local behaviour”.






        share|cite|improve this answer
























          1












          1








          1






          An ideal is prime if and only if its set-theoretical complement is multiplicatively closed. (We want to keep obtaining non-multiples of $p$ when multiplying non-multiples of $p$.)



          This allows for constructing a new ring (addition and multiplication) of “fractions” whose “denominators” live only in the complement of the prime ideal (such construction is called “localization”). This new ring is simpler than the original ring, exposing only “local behaviour”.






          share|cite|improve this answer












          An ideal is prime if and only if its set-theoretical complement is multiplicatively closed. (We want to keep obtaining non-multiples of $p$ when multiplying non-multiples of $p$.)



          This allows for constructing a new ring (addition and multiplication) of “fractions” whose “denominators” live only in the complement of the prime ideal (such construction is called “localization”). This new ring is simpler than the original ring, exposing only “local behaviour”.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 2 hours ago









          ir7ir7

          4,15311015




          4,15311015























              0














              Welcome!
              take 6 for example it is obviously not prime. We have that $6|6=2*3$ but $6nmid 2$ and $6nmid 3$.



              So if this definition is not true, that means there must be "a part" of the number hidden as a factor of $a$ and a different part hidden in $b$. Which whould of course mean that the number itshelf has more than one factor so it is not prime in the classical sence






              share|cite|improve this answer


























                0














                Welcome!
                take 6 for example it is obviously not prime. We have that $6|6=2*3$ but $6nmid 2$ and $6nmid 3$.



                So if this definition is not true, that means there must be "a part" of the number hidden as a factor of $a$ and a different part hidden in $b$. Which whould of course mean that the number itshelf has more than one factor so it is not prime in the classical sence






                share|cite|improve this answer
























                  0












                  0








                  0






                  Welcome!
                  take 6 for example it is obviously not prime. We have that $6|6=2*3$ but $6nmid 2$ and $6nmid 3$.



                  So if this definition is not true, that means there must be "a part" of the number hidden as a factor of $a$ and a different part hidden in $b$. Which whould of course mean that the number itshelf has more than one factor so it is not prime in the classical sence






                  share|cite|improve this answer












                  Welcome!
                  take 6 for example it is obviously not prime. We have that $6|6=2*3$ but $6nmid 2$ and $6nmid 3$.



                  So if this definition is not true, that means there must be "a part" of the number hidden as a factor of $a$ and a different part hidden in $b$. Which whould of course mean that the number itshelf has more than one factor so it is not prime in the classical sence







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 3 hours ago









                  George NtouliosGeorge Ntoulios

                  557




                  557






















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