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 R­module and S = EndR (M) is semiperfect. Show that M is a finite direct sum of strongly indecomposable submodules M i . (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 {x i } i#I # P of elements in P and a family {f i } i#I # HomR (P, R) of homomorphisms such that: (i) for each x # P there are only finitely many homomorphisms f i with f i (x) #= 0; and (ii) for each x # P we have x = # i#I f i (x)x i . (Such a pair of families is called a dual basis for P . Thus P is projective if and only if P has Collections: Mathematics