Rings and Algebras Problem set #10. Nov. 24, 2011. 1. Let R be a semiperfect ring with a left ideal I. Suppose J(R) is nil. Show that I contains a Summary: Rings and Algebras Problem set #10. Nov. 24, 2011. 1. Let R be a semiperfect ring with a left ideal I. Suppose J(R) is nil. Show that I contains a nonzero idempotent element whenever I is not contained in J(R). 2. Prove that an integral domain is semiperfect if and only if it is a local ring. 3. Prove that a commutative ring is semiperfect if and only if it is a finite direct product of local rings. 4. Suppose M is an R-module and S = EndR(M) is semiperfect. Show that M is a finite direct sum of strongly indecomposable submodules Mi. (Note that the converse also holds: if M is a direct sum of finitely many strongly indecomposable modules then the endomorphism ring of M is semiperfect.) 5. Show that for a module P the following are equivalent: (i) P is projective (i. e. the functor HomR(P, -) is exact); (ii) if M/K P for some module M and submodule K then K is a direct summand of M; (iii) there exists a module P such that P P is a free module; (iv) there exists a free module F such that P F is a free module. 6. Let P be an R module. Show that P is projective if and only if there exists a family {xi}iI P of elements in P and a family {fi}iI HomR(P, R) of homomorphisms such that: (i) for each x P there are only finitely many homomorphisms fi with fi(x) = 0; and (ii) for each x P we have x = iI fi(x)xi. (Such a pair of families is called a dual basis for P. Thus P is projective if and only if P has Collections: Mathematics