Injectivity in some categories of semimodules over semirings

Abstract

A semiring is a weakened version of a ring in which additive inverses are not assumed to exist. One can define semimodules over semirings by analogy with the way one defines modules over rings, and, by copying the definition of injective modules almost verbatim, injective semimodules. Unfortunately, many theorems about injective modules do not seem to be true for injective semimodules if this definition is used, which suggests a different definition might be better. In this research and alternative definition is proposed, one which allows several of the fundamental theorems from the module case to be proved for semimodules as well. Let S be a fixed semiring, and let Cs denote the category of all cancellative S-semimodules. An S-semimodule I is Cs-injective iff Iϵ Cs and for each pair of elements A and B of Cs, each S-monomorphism f:A→B, and each S-homomorphism g:A→I, there exists and S-homomorphism h:B→I such that g=h·f. An S-semimodule B is and essential extension of a subsemimodule A iff for every S-semimodule C and every S-homomorphism f:B→ C, fla is injective implies f is injective. Finally, an S-semimodule I is a Cs-injective hull of an S-semimodule A iff I is Cs-injective and I is and essential extension A. The main results of this research are as follows: (i) every cancellative S-semimodule is a subsemimodule of a Cs-injective semimodule, consequently if there exists a nonzero cancellative S-semimodule, then there exists a nonzero Cs-injective semimodule; (ii) an S-semimodule is Cs-injective iff it has no proper cancellative essential extension; and (iii) every cancellative S-semimodule had a unique Cs-injective hull.