MATH 611 Due Wednesday November 25, 1992

1. Let M be a left R-module. Show that the following conditions are equivalent: (1) Every left invertible φ in End_{R} M is also right invertible.

(2) Every right invertible endomorphism is also left invertible.

(3) No proper summand of M is isomorphic to M.

2. a) Prove that a module with finite length over any ring is finitely generated. b) Prove that if R is an artinian ring without zero divisors then R is a skew field.

c) Prove that if p is a prime ideal in a commutative ring R and R/p is an artinian R-module then p is maximal.

d) Prove that if M is a non-trivial module with finite length over a commutative ring R, then Ass M is not empty and consists of maximal ideals.

3. Let M = K ⊕ N be a direct sum of R-modules.

a) Prove that if L is a submodule of M such that N ⊆ L, then L = N ⊕ (L ∩ K).

b) A submodule H of M is called fully invariant if φ(H) ⊆ H for all φ ∈ End_{R} M. Prove that if H is a fully invariant submodule of M then H = (H ∩ K) ⊕ (H ∩ N).

c) Prove that if H is a direct summand of M (i. e. M = H ⊕ H′ for some H′) and H is fully invariant in M, then H ∩ N is a direct summand of N.

4. Let L be a minimal (non-trivial) left ideal in a ring R and suppose that at least one maximal left ideal of R does not contain L.

a) Prove that L is a direct summand of the left R-module R.

b) Prove that L is generated by an idempotent (i. e. by an element e such that e^{2} = e).

5. Let L be a minimal (non-trivial) left ideal in a ring R and suppose that there exists x ∈ L such that Lx ≠ 0. Prove that L is generated by an idempotent. (hint: Prove that the map φ ∈ End_{R} L given by ℓ ↦ ℓx is an automorphism of L.)