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