Principal Ideals and The Correspondence Theorem for Rings

abstract-algebraidealsring-theory

Let $R$ be a ring and $I\trianglelefteq R$ its ideal. Then we know by the Correspondence Theorem for Rings that there is a bijection
\begin{align*}
\Phi:\{J\trianglelefteq R : \ I\subseteq J\} \longrightarrow \{K \trianglelefteq R/I \}, \quad
J \longmapsto \Phi(J):=J/I.
\end{align*}
My questions are the following.

  1. Can we claim that: $J$ is principal $\iff J/I$ is principal? For example lets take $I:=\langle X^2,Y^2 \rangle \trianglelefteq \Bbb Z_5[X,Y]$ and $J:=\langle X , Y\rangle \supseteq I$. We know that $J$ is not principal in $R$, so can we say that $J/I \trianglelefteq R/I$ is also not principal in $R/I$?

  2. A similar question will be: Lets assume that $S_1,S_2$ are rings, $J_1 \trianglelefteq S_1$ and $\varphi:S_1 \longrightarrow S_2$ is a ring epimorphism. Then, if $J_1$ is principal, so does $\varphi(J_1)$. Right?

Best Answer

Using your notation, if $J$ is generated by a single element $a$, then $J/I$ must be generated by the image of $a$ in $J/I$. The converse doesn't hold, however. Consider $(2, x) \subset \mathbb Z[x]$ and $I = (x)$.

(2) follows from the first part above since $S_2$ is isomorphic to a quotient of $S_1$.

As for your example, note that the quotient ring in the example is a finite ring; its elements have the form $a + bx + cy + dxy$ where the coefficients are in $\mathbb Z/(5)$. It can be directly checked that no such element can generate $(x, y)$.

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