Finitely Generated Modules over Semilocal Rings and Characterizations of (Semi-)Perfect Rings

Abstract

Lomp (9) has studied finitely generated projective modules over semilocal rings. He obtained the following: finitely generated projective modules over semilocal rings are semilocal. We shall give necessary and sucient conditions for finitely generated modules to be semilocal modules. By using a lifting property, we also give characterizations of right perfect (semiperfect) rings. Our main results can be summarized as follows: (1) Let M be a finitely generated module. Then M has finite hollow dimension if and only if M is weakly supplemented if and only if M is semilocal. (2) A ring R is right perfect if and only if every flat right R-module is lifting and every right R-module has a flat cover if and only if every quasi-projective right R-module is lifting. (3) A ring R is semiperfect if and only if every finitely generated flat right R-module is lifting if and only if RR satisfies the lifting property for simple factor modules. In this note, all rings R considered are associative rings with identity and all modules are unital right R-modules unless indicated otherwise. For a module M, Rad(M), Soc(M), E(M), EndR(M) are the (Jacobson) radical, socle, injective hull and endomorphism ring of M, respectively. Let M be a module and let K be a submodule of M. K is called small submodule (or superfluous submodule) of M, abbreviated K M, if, for every submodule L M, K + L = M implies L = M. Let N1 N2 M. N1 is a co-essential submodule of N2 in M, abbreviated N1 c N2 in M, if N2/N1 M/N1. A submodule N of M is said to be co-closed in M (or a co-closed submodule of M), if N has no proper co-essential submodule in M. i.e., N 0 c N in M implies N = N 0 . Let N1 N2 M. N1 is said to be a co-closure of N2 in M if N1 is a co-closed submodule of M with N1 c N2 in M. Any submodule of a module has a closure. However, a co-closure does not exist in general, for example, 2Z does not have a co-closure in ZZ. Let M be a module and let N and L be submodules of M. N is called a supplement of L if M = N + L and N L N. Note that any supplement