Some theorems in skew-semimodules over semirings

Abstract

A skew-semimodule M over semiring S is an additive monoid M with a left action SxM M, defined by (s,m) -> sm, such that for all r,s [is an element of] S and m,n [is an element of] M (i) (r+s)m = rm+sm, (ii) s(m+n) = sm+sn, (iii) (rs)m=r(sm) and (iv) s0=0 where 0 is the identity of M. A non-empty subset A of a skew-semimodule M over a semiring S is said to be an ideal of M if A+M,M+A and S* A are subsets of A where S* = S\{0}. Moreover, given an ideal A of M, the Rees congruence on M generated by A is the congruence relation R[subscript A] = {(m,n) [is an element of] MxM = n or m, n [is an element of] A}. Let M and N be skew-semimodules over a semiring S. A mapping [phi] : M -> N is called a homomorphism if (i) [phi](m+ n) = [phi](m) + [phi](n), (ii) [phi](sm) = s[phi](m) and (iii) [phi](0) = 0 for all m,n [is an element of] M and s [is an element of] S. The set of m [is an element of] M such that [phi](m) = 0 is called the zero set of [phi], denoted by Zs[phi]. In addition, the kernel of [phi] is the relation Ker[phi] = {(m, n) [is an element of] MxM |[phi](m) = [phi](n)}. Let M, N and P be groups and skew-semimodules over a semiring S. A sequence M f-> N g-> P of skew-semimodules and homomorphisms is said to be exact at N if Imf = Zsg. A chain A1 [is less than or equal to] A2 [is less than or equal to] ... or A[subscript 1] [is more than or equal to] A[subscript 2] [is more than or equal to] ... of subsets of a skew-semimodule M over a semiring S is said to be an ideal series of M if Ai is an ideal of M for all positive integers i. The main purpose of this research is to generalize of Isomorphism Theorems, the universal mapping properties of direct products, direct sums and free modules, some theorems of exact sequences and Artinian and Noetherian modules to those of skew-semimodules