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1992 Nov

1992 November

  1. a. Define ``injective module.'' b. Prove: The module \(Q\) is injective if every diagram with exact row

    (missing diagram)

    with projective \(P\) is embeddable in a commutative diagram

    (missing diagram)

  2. 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).)

  1. 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 \(i\geq n\).

  2. 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 \(M = K \oplus (1_M + \varphi)(L)\).

  3. 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.

  4. 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 \(\ker(\varphi) = (x-\alpha, y-\beta)\).

  5. 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 \(R\).

  6. 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 \(R\).