Projectivity and injectivity in semimodules

Abstract

It is the purpose of this paper to generalize the concepts of projectivity and injectivity to semimodules (commutative. cancellative semigroups with additive identity) over a hemiring A. A hemiring is a semiring with commutative addition and additive identity. All semirings in this paper are assumed to be hemirings. The first chapter deals with the necessary preliminary results including the theorem that a semimaximal homomorphism is a monomorphism if and only if it has trivial kernel. This is false for semigroup homomorphisms in general. In Chapter II the notion of a short exact sequence and a split exact sequence is developed and the preservation of exactness in the dual sequence of homomorphisms is studied. In the third chapter projective and semiprojective semimodules are defined and it is shown that every free semimodule is semiprojective, but not conversely. Also, some basic results on free semimodules are established and it is shown that there does not exist a semisubtractive free semimodule over a hemiring with cancellative addition th.at has more than one generator. The last chapter deals with injective semimodules. A semimodule Mover a hemiring A is said to be injective if for every k-ideal of A and every A-homomorphism g:I to M, there exists an extension to all of A.. It is shown that if A is a hemiring such that every ideal is a k-ideal, then every semimodule M in a certain class is a subsemimodule of an injective A semimodule. Next the notion of Sigma injectivity is generalized to semimodules and finally semimodules over domains are investigated. It is proved that a divisible semimodule over a principal k-ideal domain A is injective. An example of a semisubtractive principal ideal hemiring is given.