Reduction of a family of ideals

In the paper we prove that there exists a simultaneous reduction of one-parameter family of $\mathfrak{m}_{n}$-primary ideals in the ring of germs of holomorphic functions. As a corollary we generalize the result of A. P\l{}oski \cite{ploski} on the semicontinuity of the \L{}ojasiewicz exponent in a multiplicity-constant deformation.