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.
