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