On a finite dimensional quasi-simple module

Abstract

1. If R is a ring and M is a (right) A'-module such that MR^O, then M is said to be quasi-simple provided that (Ql) the endomorphism (module endomorphism) ring of the quasiinjective hull of M is a division ring, (Q2) if N is a nonzero submodule of M then there is a nonzero endomorphism / of M such that /(M) C N. Any simple module is clearly quasi-simple, however a quasi-simple module is not necessarily simple. For example, if R is a semiprime ring with a uniform right ideal U such that the right singular ideal of R is zero then the regular A'-module U is quasi-simple since the endomorphism ring of its quasi-injective hull is a division ring [5, 1.7, p. 263] and for any nonzero submodule N oi U there is xEN such that O^acf/CJV. Let us say that a quasi-simple module is finite dimensional if its quasi-injective hull is a finite dimensional vector space over its endomorphism ring. The main results in this paper are the following: Let ¥ be a faithful quasi-simple A'-module for some ring R and let M be the quasi-injective hull of M. Let D = Hom/j(iif, M) and if N is a nonzero submodule of M then define K(N) = Ylomn(N, N). Then for any nonzero submodule N of M, K(N) is a right order in D, M=DN and if {7wt-}=1 is a finite sequence of Z?-linearly independent elements in M and if [y}=1 is a finite sequence in M, then there exist rER, 0^&GHomfi(Af, M) such that 0^k\NEK(N) and mtr = kyi for all l^if$n. If, in addition, the singular submodule of M is zero then, for any large right ideal B of R one can choose rG-B and 0^i£HomR(M, M) such that 0^k\N EK(N) and mir = kyt for all l^ijSw. In case the set {mi, m2, • • • , mn is a basis for M then r is a regular element. If R is a right Goldie prime ring and U is a uniform right ideal then U is a finite dimensional quasi-simple module. Conversely, if A* has a faithful finite dimensional quasi-simple right module M then A3 is a right Goldie prime ring, and M is isomorphic to a uniform right ideal of R. Hence our theory above provides a new proof for Theorem 10 of [3, p. 603]. Let Dn be the nXn matrix ring representing r\omD(M, M) relative to the basis {mi, m2, • ■ ■ , mn] and K(N)n be the nXn matrix ring over K(N). Let R* be the subring of Dn which represents R as a ring of linear transformations on M over D relative to the same basis.