Injective property of Laurent power series module (Q2785288)

Let \(R\) be an associative ring, \(\mathbb{N}\) be the set of all natural numbers and let \(\mathbb{Z}\) be the set of all integers. Let \(S=\{0,k_1,k_2,k_3,\dots\}\) be a submonoid of \(\mathbb{N}\) and let \(M\) be a left \(R\)-module. The Laurent power series, denoted by \(M[[x^{-1},x]]=\{\sum_{i\in\mathbb{Z}}m_ix^i\mid m_i\in R,\;\forall i\in\mathbb{Z}\}\) is a left \(R[x^S]\) module such that \(r\sum_{i\in\mathbb{Z}}m_ix^i=\sum_{i\in\mathbb{Z}}rm_ix^i\), where \(r\in R\) and \(x^{k_j}\sum_{i\in\mathbb{Z}}m_ix^i=\sum_{i\in\mathbb{Z}}m_ix^{i+k_j}\), where \(k_j\in S\) for every \(j\in\mathbb{N}\). The main result established by the author is: If \(E\) is an injective left \(R\)-module, then the Laurent power series \(E[[x^{-1},x]]\) is an injective left \(R[x^S]\)-module.