 
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
