PROPERTIES OF INDUCED INVERSE POLYNOMIAL MODULES OVER A SUBMONOID

Abstract

Let M be a left R-module and R be a ring with unity, and S = {0,2,3,4,···} be a submonoid. Then M[x^{−s}] = {a_0 + a_2x^{−2}+a_3x^{−3}+···+a_nx^{−n} |a_i ∈M} is an R[x^s]-module. Inthis paper we show some properties of M[x−s] as an R[xs]-module.

Let f : M → N be an R-linear map and \bar{M}[x^{−s}] = {a_2x^{−2} + a_3x^{−3} +···+a_nx^{−n} | a_i ∈ M} and define N +M[x^{−s}] = {b_0 + a_2x^{−2} +a_3x^{−3} +···+a_nx^{−n} | b_0 ∈ N, a_i ∈ M}. Then N +M[x^{−s}] is an R[x^s]-module.

We show that given a short exact sequence 0 → L → M → N → 0 of R-modules, 0 → L → M[x^{−s}] → N+ \bar{M}[x^{−s}] → 0 is a short exact sequence of R[x^s]-module. Then we show E_1 + \bar{E_0}[x^{−s}] is not an injective left R[x^s]-module, in general.