Local Units Semigroup Graded Rings Of Alluvial And Roots

Let Sbe a local unit semigroup, R be a S-graded ring. In this paper, we define the Smash Product R#S* of S-graded ring on the basis of Smash Product R#K[Gf]*. Then we prove that R#S* is an associative ring and study the equivalence between the category R#S*-Mod and the graded (S,R)-gr. Moreover, we discuss the relation of the related radicals and obtain the results of Jacobson radical of the ring R#S*. J(R#S*) = Js(R)#S* and Jref(R) = (J(R#S*))↓= JS(R).At last, we discuss the local unit properties of Smash Product R#S* of 5-graded ring in case that S is an infinite semigroup and generalize some previous results.