Generalization of some theorems in module theory to skewmodules

Abstract

Let R be a skewring. An R-skewmodule M is an additive group with a left action RxM -> M, defined by (r,m) -> rm, such that (1) (r+s)m = rm+sm, (2) r(m+n) = rm+rn and (3) (rs)m = r(sm) for all r,s R and m,n M. A subgroup N of an R-skewmodule M is called a subskewmodule of M if for all n N and r R, then rn N. Moreover, N is called a normal subskewmodule if N is a subskewmodule of M such that N+m = m+N for all m M. An R-skewmodule M is simple if {0} and M are only normal subskewmodules of M. Let M be an R-skewmodule. Normal subskewmodules M1 and M2 of M are said to be supplementary if M = M1+M2. A normal subskewmodule N of M is called a direct summand if there exists a normal subskewmodule P of M such that N and P are supplementary. The main results of this research are follows: Generalization the notion of the four Isomorphism Theorems, the schreier's theorem and the Jordan Holder theorem in module theory to skewmodules. Moreover we obtain the following theorems: Theorem1 Let M be an R-skewmodule. If M is both artinian and noetherian, then M has a composition series. Theorem2 Let M be an R-skewmodule. If M is the sum of a family of its normal simple subskewmodules, then every normal subskewmodule of M is a direct summand.