Irreducible multiplicity of idealizations

Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitely generated $R$-module. We study the relations of the index of reducibility and the irreducible multiplicity of an $\mathfrak{m}$-primary ideal of $R$ and these of $\mathfrak{m}\times M$-primary ideal of the idealization. This generalizes one of the main results of S.Goto et al. (see \cite[Theorem 2.2]{GKL}).