The category of supermodules

Given a commutative algebra with unit element R and an R -module U, in this paper we will construct a category consists as objects all pairs of R -modules (N,M) where N is direct sum of U and M contains U . To define the morphisms between any two objects, we will identify an object (N,M) with an R -module E determined by N and M . By this method, we obtain a category denoted by CU. Further, if W is a direct summand of U then there exists a functor from CU to CW.


Introduction
Given a commutative algebra with unit element R and an R -module U , our goal is to construct a category C of supermodules of U whose objects are all pairs of R -modules ( ) , NM where N is direct sum of U and M contains U . To define the morphisms between any two objects in C , we will identify every object ( ) , NM uniquely with an R -module E determined from N and M . We will call such category as "supermodule category of U ". The idea of this research was generated from the notion of U -exact sequence introduced by Davvaz and Parnian-Garamaleky in [1] and by Davvaz and Shabani-Solt in [2]. Mahatma and Muchtadi Alamsyah then developed the U -extension module in [3] and, further, Mahatma investigated the equivalence between the first U -extension module and the short U -exact sequence [4]. The last manuscript gives a method for determining a pair of supermodule with a module. This research can be continued by further question about the existence of equivalence between two categories of supermodules, either generated using the same algebra or different algebras.

Method of research
The category C we construct in this article whose objects are all pairs of R -modules (  2 We note first that when defining the module E of ( ) , NM , we must involve the submodule U in the process. For if not then the category of supermodules of U will be the same with the category of supermodules of W for any submodule W of U .

Constructing the category Let
R be commutative algebra with unit element and U be projective We will show that the module E does not depend on the choice of z . Suppose that : This implies Clearly,  is an by  . Therefore,  is an isomorphism. We call this isomorphism as "standard isomorphism". Notice that the standard isomorphism : EE  → is actually the identity map 1 E .
We have shown that every ( ) We define the composition of two morphisms in C as follows: Suppose Proof: We only need to show the associativity of the composition and the existence of the identity morphism for every object in C . First suppose that 1 : Notice that if

Result and discussion
We have given a method for constructing a category of supermodules of an R -module U , called It is not hard to verify that if U and W are isomorphic then F become a category equivalence. There is still a question about what the necessary condition is for the existence of equivalence between two categories. It will be interesting to investigate further whether there is an equivalence between two categories of supermodules over different algebras.

Conclusion
Given commutative algebra with unit element R and an R -module U there exists a category of supermodules of U consists of all pairs of R -modules ( ) , N U V M U =   as objects. Further, if W is direct sum of U then there exists a functor from the category of supermodules of U to the category of supermodules of W .