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

• Published in: Journal of physics. Conference series 2018-09, Vol.1097 (1), p.12078
• Main Authors: Hijriati, N, Wahyuni, S, Wijayanti, I E
• Format: Article
• Language: English
• Subjects: Homomorphisms; Modules; Physics; Representations
• ISSN: 1742-6588; 1742-6596
• DOI: 10.1088/1742-6596/1097/1/012078

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