Injectivity and Projectivity Properties of The Category of Representation Modules of Rings

Abstract

Let R, S be two rings with unity, M an S-module, and f: R → S a ring homomorphism. If the map M → M, m ↦ f(r)m is S-linear for any r ∈ R, then M is a representation module of ring R. This condition will be true if sf (r) − f (r)s ∈ Ann(M) for all r ∈ R and s ∈ S. The class of S-modules M, where sf(r) − f(r)s ∈ Ann(M) for all r ∈ R and s ∈ S, forms a category with its morphisms are all module homomorphisms. This class is denoted by \U0001d50d. The purpose of this paper is to prove that the category \U0001d50d is an abelian category which is under sufficient conditions enabling the category \U0001d50d has enough injective objects and enough projective objects. First, we prove the category \U0001d50d is stable under kernel and image of module homomorphisms, and a finite direct sum of objects of \U0001d50d is also the object of \U0001d50d. By using this two properties, we prove that \U0001d50d is the abelian category. Next, we determine the properties of the abelian category \U0001d50d, such that it has enough injective objects and enough projective objects. We obtain that, if S as R-module is an element of \U0001d50d, then the category \U0001d50d has enough projective objects and enough injective objects.