Hilbert-Kunz Multiplicity of Fiber Product Rings and Nagata Idealizations

The main purpose of this paper is to provide formulas for the Hilbert-Kunz multiplicity of fiber product rings and Nagata idealizations. We give explicit formulas for the Hilbert-Kunz multiplicity of a fiber product $R \times_T S$, where $R$, $S$, and $T$ are Noetherian local rings sharing the same characteristic and residue field. We compute the Hilbert-Kunz multiplicity of a general Nagata idealization $R \ltimes M$, where $M$ is a finitely generated $R$-module. Additionally, we provide examples, structural results and establish new bounds for the Hilbert-Kunz multiplicity.