Exact category of hypermodules

The theory of hyperstructures has been introduced byMarty in 1934 during the 8th Congress of the Scandinavian Mathematicians [4]. Marty introduced the notion of a hypergroup and since then many researchers have worked on this new topic of modern algebra and developed it. The notion of a hyperfield and a hyperring was studied first by Krasner [2] and then some authors followed him, for example, see [8]. The canonical hypergroups are a special type of hypergroup. Initially they were derived from the additive part of the hyperfield and hyperring. The name canonical has been given to these hypergroups by Mittas, who is the first one that studied them extensively [7]. Again in the context of canonical hypergroups some mathematicians, for example, [5] studied hypermodules whose additive structure is just a canonical hypergroup. Considering the class of hypermodules over a fixed hyperring R and the class of all homomorphisms among hypermodules together with the composition of the mappings, knowing that the composite of two homomorphisms is again a homomorphism and that for any hypermodulesA over the hyperring R, idA : A→ A, id(a)= a, is a homomorphism among hypermodules, we can construct a category which is denoted by Hmod. In this paper some aspects of hypermodules are studied. We will show that the category Hmod is exact in the sense that it is normal and conormal with kernels and cokernels in which every arrow f has a factorization f = νq, with ν being a monomorphism and q an epimorphism (see [6]). Two of the most used results of the paper are those which state that the monomorphisms of Hmod (in the categorical sense) are the oneto one-homomorphisms and the epimorphisms of Hmod are the onto homomorphisms.