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I'm trying to show that $x$ and $y$ are irreducible and non-associate elements of $R := \mathbb{Z}[x,y]/(xy).$

To show they're non-associate elements, suppose $x=uy$ for some unit $u\in R.$ Then there exists $v\in R$ so that $uv = 1.$ Also, $x=uy\Rightarrow x-uy \in (xy)\Rightarrow x-uy=(f)(xy)$ for some $f \in \mathbb{Z}[x,y].$ But this isn't possible as $(f)(xy)$ cannot contain the term $x$.

Now suppose $x=ab$ for some $a, b\in R.$ I need to show that either $a$ or $b$ is a unit. Maybe some isomorphism mapping $x$ and $y$ to particular elements might be useful?

Is the proof that $x$ is not associates with $y$ correct? How do I show that $x$ and $y$ are irreducible?

Edit: I've decided to make some clarifications on irreducible and nonassociate elements. Also, just in case, $(xy)$ is the smallest ideal of $\mathbb{Z}[x,y]$ containing $xy.$ Let $R$ be a ring. An irreducible element $x$ is a nonzero nonunit such that if $x = ab$ for some $a,b \in R,$ then one of $a$ or $b$ is a unit (e.g. if $a$ is a unit, then there exists $v \in R$ so that $av = va = 1_R$). Two elements $a,b\in R$ are associates if $a = ub$ for some unit $u.$

user3472
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  • Note $x=x(1+y)$. – Gerry Myerson Jul 29 '21 at 02:49
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    You should define "associate" and "irreducible" since there are no standard definitions in rings with zero-divisors, cf. here. – Bill Dubuque Jul 29 '21 at 10:02
  • Note : I wrote an answer, but I've deleted it because I'm not aware if the notions expressed above match with those I used below. You may attach the name of the textbook that you usually refer to while reading up these concepts (or any standard source that would use the same definition as you) and I'll be able to help you out, possibly even explaning what would happen if the other definition was used. – Sarvesh Ravichandran Iyer Jul 29 '21 at 10:20
  • @TeresaLisbon I've clarified the definitions. Hope that helps. – user3472 Jul 29 '21 at 12:32
  • @user3472 Thanks very much for your proactivity (I gave a +1 earlier). I will confirm that our definitions match, and see if any issues crop up. I had written an answer, but once Bill pointed the issue out I deleted my answer fearing that it may confuse you. It's the "danger" of reading from a single textbook, though. A bit like limit points having various names/definitions in various textbooks, it feels like there are a couple of sticky points here and there in algebra as well, and I'll see if I can provide you with a link of conflicting algebraic notions sometime. – Sarvesh Ravichandran Iyer Jul 29 '21 at 12:56

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