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$\color{black}{Background:}$

$\textbf{Definition:}$

If $R$ is an integral domain, $p\in R$ is called an $\textbf{irreducible element}$ and is said to be $\textbf{irreducible}$ in $R$) if it satisfies the following conditions:

$(1)$ $p\neq 0$ and $p$ is not a unit

$(2)$ if $p=ab$ in $R,$ then $a$ or $b$ is a unit in $R.$

An element that is not irreducible is called $\textbf{reducible.}$

$\color{black}{Questions:}$

What I would like to know is in the above definition for irreducible element in an integral domain $R,$ if $p=ab$ and only one of $a,b$ is a unit, say $b$ is a unit, and $a$ is not a unit. Then $a$ is an associate of $p.$ But does $a$ has to be a prime or non prime element, another irreducible element, or can be neither of the two. I know that as an assocaite to $p$ $a$ is call a unit multiple $p.$ To be precise, what I am asking is, does every associate of an irreducible element can be assigned some sort of classififcation or a name for them.

Say in $Z_6$, we have $2\equiv 2\cdot 3 \pmod 6$, $2\equiv 4\cdot 5 \pmod 6$, $2,4$ are associates of $2,$ but since $4$ is not a unit, but can it be something that is neither prime, irreducible, or not fall into any kind of classification other than just being an associate to $2$.

Thank you in advance

Seth
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  • Divisibility theory is much more complex in rings with zero-divisors. In particular there is no standard definition of "associate", "unique factorization", etc. See the linked dupe for literature citations (and see the Linked questions there). – Bill Dubuque Jan 24 '24 at 19:12
  • @BillDubuque the reason I ask is that I have look at many abstract algebra textbooks, and I did the exercises about associates, but it seems like not much is said about if other than not being a unit, can they always have some sort of classification according to whatever elements of commutative rings can have, like being prime, units, irreducible,,etc. Also, it felt like everytime the notion of associate get introduced, the authors just left it hanging. Why? Well, in th e section on UFD, PID and ED,, – Seth Jan 24 '24 at 19:15
  • @BillDubuque it seems that every element of a commutative get some sort of description or name based on some sort of properties. Assicoates seem to get the short end of the stick and it feel like it got left hanging. – Seth Jan 24 '24 at 19:21
  • If $p$ and $q$ are associate then $p$ is prime $\iff q$ is prime. In many divisibility problems units play no role so we ignore units, i.e we work up to associates, as in the statement of uniqueness of prime factorizations. – Bill Dubuque Jan 24 '24 at 19:21
  • @BillDubuque thank you for the reference post. I will take a look at it. If I have any more questions, I will leave them here for you if that is okay with you. – Seth Jan 24 '24 at 19:26
  • In domains the associates are exactly the unit multiples so studying associates amount to studying the group of units. But there is no universal classification of associates. – Bill Dubuque Jan 24 '24 at 19:28
  • @BillDubuque "the associates are exactly the unit multiples so studying associates amount to studying the group of units" is there a theorem that captures that statement? – Seth Jan 24 '24 at 19:32
  • It's in the paper I cite in the dupe. See also here. – Bill Dubuque Jan 24 '24 at 20:11
  • @BillDubuque ah ok ok thank you. – Seth Jan 24 '24 at 20:16

0 Answers0