Some Properties of Serre Subcategories in the Graded Local Cohomology Modules

Abstract

LetR=⊕n≥0Rnbe a standard homogeneous Noetherian ring with local base ring(R0,m0)and letMbe a finitely generated gradedR-module. LetHR+i(M)be theith local cohomology module ofMwith respect toR+=⊕n>0Rn. LetSbe a Serre subcategory of the category ofR-modules and letibe a nonnegative integer. In this paper, ifdim⁡R0≤1,then we investigate some conditions under which theR-modulesR0/m0 ⊗R0 HR+i(M),Γm0R(HR+i(M))andHm0R1(HR+i(M))are inSfor alli≥0. Also, we prove that ifdim⁡R0≤2, then the gradedR-moduleHm01(HR+i(M))is inSfor alli≥0. Finally, we prove that ifnis the biggest integer such thatHai(M)∉S, thenHR+i(M)/m0HR+i(M)∈Sfor alli≥n.

On Nearly Prime Submodules of Unitary Modules

“…Modules over associative rings play important roles in the investigation of ring constructions (see [1,2]). Modules are very important and have been actively investigated (see, for example, [3][4][5][6][7]). Throughout this paper, all rings are associative with identity and all modules are unitary rightmodules.…”