Semicorings and Semicomodules

Abstract

In this paper, we introduce and investigate semicorings over associative semirings and their categories of semicomodules. Our results generalize old and recent results on corings over rings and their categories of comodules. The generalization is not straightforward and even subtle at some places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no non-zero injective objects. To overcome these and other difficulties, a combination of methods and techniques from categorical, homological and universal algebra is used including a new notion of exact sequences of semimodules over semirings.

On Semisimple Semirings

“…In addition to the categorical notions of projective and injective semimodules over a semiring, new notions of projectivity and injectivity of semimodules over semirings were considered by the first author who introduced the so called e-projective and e-injective semimodules [Abu2014-CA]. One reason for the interest in such notions is the phenomenon that assuming that all semimodules of a given semiring S are projective (injective) forces the semiring to be a ring (cf., [Ili2010, Theorem 3.4]).…”