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The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber products and idealization rings. We calculate the Hilbert-Kunz multiplicity of a fiber product R T S, with R, S and T being Noetherian local rings with common characteristic and residue field. Inspired by a work of Strazzanti and Armengou we calculate the Hilbert-Kunz multiplicity of an idealization ring R M, where M is a finite R-module. We explore certain examples and structural consequences and obtain some bounds for the Hilbert-Kunz multiplicity.
Pérez et al. (2024) studied this question.