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 $S=\{0,2,3,4,\cdots\}$ be a submonoid. Then $M[x^{-s}]=\{a_0+a_2x^{-2}+a_3x^{-3}+\cdots +a_nx^{-n} \mid a_i \in M\}$ is an $R[x^s]$-module. In this paper we show some properties of $M[x^{-s}]$ as an $R[x^s]$-module. Let $f:M \longrightarrow N$ be an $R$-linear map and $\overline{M}[x^{-s}]=\{a_2x^{-2}+a_3x^{-3}+\cdots +a_nx^{-n} \mid a_i \in M\}$ and define $N+\overline{M}[x^{-s}]=\{b_0+a_2x^{-2}+a_3x^{-3}+\cdots +a_nx^{-n}\mid b_0 \in N, a_i \in M\}$. Then $N+\overline{M}[x^{-s}]$ is an $R[x^s]$-module. We show that given a short exact sequence $0 \longrightarrow L \longrightarrow M \longrightarrow N \longrightarrow 0$ of $R$-modules, $0 \longrightarrow L \longrightarrow M[x^{-s}]\longrightarrow N+\overline{M}[x^{-s}] \longrightarrow 0$ is a short exact sequence of $R[x^s]$-module. Then we show $E_1+\overline{E_0}[x^{-s}]$ is not an injective left $R[x^s]$-module, in general. 수식