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Let $(R,\mathfrak{m})$ be a Noetherian local ring and $M$ a finitelygenerated $R$-module. In this paper, we study the relation among the Hilbertfunction, Hilbert multiplicity, index of reducibility of an$\mathfrak{m}$-primary ideal, and the irreducible multiplicity underidealization $R\ltimes M$. Some applications to Cohen-Macaulayness andCohen-Macaulay type of idealization are given.

Hilbert multiplicity and irreducible multiplicity of idealizations