Ring Theory Exam

November 1, 1992 Reading time: about 15 minutes (3017 words).

1992 November

Let $R$ be a commutative ring with identity. Recall that an ideal $P\subset R$ is said to be prime if $P\neq R$ and $ab \in P$ implies $a\in P$ or $b\in P$. Let $J\subseteq R$ be an ideal. a. Show that $J$ is prime if and only if $R/J$ is an integral domain. b. Show that $J$ is maximal if and only if $R/J$ is a field.

(This is problem 2 of the April 2008 exam; see section (#sec:2008Apr).)
Let $R$ be a principal ideal domain. Let $J_1 \subseteq J_2 \subseteq \cdots \subseteq J_i \subseteq J_{i+1}\subseteq \cdots$ be a chain of ideals in $R$. Show that there is an integer $n\geq 1$ such that $J_i = J_n$ for all $i\geq n$.

Let $R$ be a ring and $M$ a (left) $R$-module. Suppose $M = K \oplus L$ and $\varphi$ is an $R$-endomorphism of $M$ with the property that $\varphi(L)\subseteq K$. Prove that $M = K \oplus (1_M + \varphi)(L)$.
Let $R$ be any ring, $M, N, K$ submodules of some $R$-module such that $N\leq M$. Prove that the sequence $$0 \longrightarrow \frac{M\cap K}{N\cap K} \longrightarrow \frac{M}{N} \longrightarrow \frac{M+ K}{N+ K} \longrightarrow 0$$ is exact where the maps are the natural (obvious) ones.
Let $F$ be a field and let $R = F(X,Y]$ be the polynomial ring over $F$ in indeterminates $X, Y$. Let $\alpha, \beta \in F$, and let $\varphi : R\rightarrow F$ be defined by $\varphi(f(X,Y)) = f(\alpha, \beta)$ for all polynomials $f(X,Y)\in R$. Show that $\ker(\varphi) = (x-\alpha, y-\beta)$.
Let $R$ be a ring which is generated (as a (left) $R$-module) by its minimal left ideals (i.e. $R$ is semi-simple).
a. Prove that $R$ is the direct sum of a {\bf finite} number of minimal left ideals.
b. Prove that every simple $R$-module is isomorphic to a left ideal of $R$.
Let $R$ be a commutative ring. Prove that an element of $R$ is nilpotent if and only if it belongs to every prime ideal of $R$.