Semigroup Graded Rings, Morita Duality,

This paper discuss Morita Duality of semigroup-graded rings,and get a equivalence between duality functor of graded module category and bi-graded bimodule. Firstly we give the definition of semigroup bigraded R -A-bimodule, and a duality functor between (R,S) - gr and gr - (A, ), namely category of S-graded left .R-module and -graded right A-module.In the section 2, an important role is played by the subsetsof , where . we get some useful characters of right A-module, and find an index set (q), q = 1-1A, is a system of finitely generated projec-tive generators of gr - (A, ) for some -graded rings. Results,we give the definition of semigroup gr-faithfully balanced.At the base of these, in section 3 we investigate the relationship between bigraded bimodule and that duality functor : let , be full subcategories of (R, S) - gr and gr - (A, ft) respectively, F : and F1 : - be a duality, assume that for every s S, sp , then exist a bigraded R - A-bimodule Q = sQ, such that F' isomorphic to HA(-,Q). At the sametime we abroad some results of [1]. By introducing the concept of [3] aboutgr-linearly compact discrete (gr-l.c.d.),we have proved Density Theorem in semigroup graded module, and gain a equivalence : for , SQA is gr-injection if and only if for , RQ is gr-linearly compact discrete(gr-l.c.d.).