Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 25 1 2026 07 01 A note on the first nonzero Fitting ideal of a module 106169 10.48308/cgasa.2025.236826.1518 EN Somayeh Hadjirezaei Department of Mathematics, Vali-e-Asr University of Rafsanjan, Iran 0000-0002-8994-5523 Journal Article 2024 09 07 Let $R$ be a commutative ring and $M$ be a finitely generated $R$-module. Let I$(M)$ be the first nonzero Fitting ideal of $M$. In this paper we characterize some modules over Noetherian UFDs, whose first nonzero Fitting ideal is a prime ideal. We show that if $P$ is a prime ideal and $M$ is a finitely generated R-module with I$(M) = P$ and T$(M_P)\neq 0$, then M is isomorphic to $R/P \oplus N$, for some projective R-module $N$ of constant rank. Also, we investigate some conditions under which ${M}/$T$(M)$ is free. https://cgasa.sbu.ac.ir/article_106169_4fbb34893d933629fe7d7e31ed3d7cc0.pdf