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 �: � → � 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 ℑ. The purpose of this paper is to prove that the category ℑ is an abelian category which is under sufficient conditions enabling the category ℑ has enough injective objects and enough projective objects. First, we prove the category ℑ is stable under kernel and image of module homomorphisms, and a finite direct sum of objects of ℑ is also the object of ℑ . By using this two properties, we prove that ℑ is the abelian category. Next, we determine the properties of the abelian category ℑ, such that it has enough injective objects and enough projective objects. We obtain that, if S as R-module is an element of ℑ, then the category ℑ has enough projective objects and enough injective objects.