| Abstrak/Abstract |
The representation of rings on finite dimension vector spaces has been generalized to the representation of rings on modules over a commutative ring. Let S be a commutative ring with unity and M an S-module. A representation of ring R with unity on an S-module M is a ring homomorphism from R to the ring of endomorphisms of M. An S-module associated with a representation of R is called a representation module of R. For any ring homomorphism f : R → S, we define a representation of ring R with unity on M via f, and it is called an f-representation of ring R which is a special case of the representation of ring R on an S-module. This S-module associated with the f-representation of ring R is called an f-representation module of R.
In case S is non-commutative, we give a sufficient condition for the S-module M to be a representation module of R. The category of f-representation modules of ring R is Abelian and Morita equivalent to the category of modules over an R-algebra. Thus, if the category of modules over the R-algebra which is equivalent to the category of f-representation modules of R satisfies the Krull-Schmidt Theorem, then the category of f-representation modules of R also satisfies Krull-Schmidt’s Theorem.
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