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I am studying by myself and I needed help for few question which I am confused how give proof of that. Let $\varphi : J \to K$ be a ring epimorphism with $\varphi(1) = 1$, where $J$ and $K$ are commutative rings with $1$. Prove the following or give a choice of $J$, $K$, and $\varphi$ where the claim fails.

  1. If $J$ is an integral domain, then $\varphi(J)$ is an integral domain.
  2. If $(k) \unlhd K$ is a principal ideal, i. e., generated by a single element, then the preimage $\varphi^{-1}((k))$ is a principal ideal in $J$.

Help me to understand to how to solve this question?

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  • $\begingroup$ For your second question, what is $S$? Also, do you mean $\varphi^{-1}[(s)]$? $\endgroup$ – Hayden Mar 31 '14 at 17:37
  • $\begingroup$ Also, what is $R$? You keep on introducing new objects without indication of what they are $\endgroup$ – Hayden Mar 31 '14 at 17:39
  • $\begingroup$ I have updated my question. $\endgroup$ – user1413 Mar 31 '14 at 17:41
  • $\begingroup$ When you work in the category of (commutative) unital rings you don't have to mention $\varphi(1) = 1$, that is part of the definition. Is $\varphi$ really just an epimorphism in the categorical sense or do you mean a surjective homomorphism? $\endgroup$ – Dune Mar 31 '14 at 17:48
  • $\begingroup$ Yes I do mean Surjective Homomorphism!! $\endgroup$ – user1413 Mar 31 '14 at 17:58
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I will assume that $\varphi: J\rightarrow K$ because of the mixed notation.

For the first question, suppose $J=\mathbb{Z}$ and $K=\mathbb{Z}/m\mathbb{Z}$, where $m$ is not prime, and $\varphi: J\rightarrow K$ is the canonical quotient transformation (which is surjective and thus epic). $\mathbb{Z}/m\mathbb{Z}$ is not an integral domain; if $m=ab$, then $ab\equiv 0 \mod m$ but $a,b$ are non-zero modulo $m$.

For the second question, take $J=\mathbb{F}[x,y]$ and $K=\mathbb{F}$. Then take $\varphi:\mathbb{F}[x,y]\rightarrow \mathbb{F}$ such that $\varphi$ is constant on constant polynomials and sends $x$ and $y$ to 0. The kernel of this is the (non-principal ideal) $(x,y)$. Thus, we take $k=0$, so that $(0)=\{0\}=\varphi(\ker \varphi)$ and thus $\varphi^{-1}[(0)]=\ker\varphi$ is not a principal ideal.

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  • $\begingroup$ So therefore $\varphi(J)$ is not integral domain. right? $\endgroup$ – user1413 Mar 31 '14 at 17:54
  • $\begingroup$ Yep, it isn't required to be an integral domain $\endgroup$ – Hayden Mar 31 '14 at 18:01
  • $\begingroup$ Oh ok got it. So the proof that I thought was right. Appreciate all for your help. $\endgroup$ – user1413 Mar 31 '14 at 18:05

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