On Finitely Generated Module Whose First Nonzero Fitting Ideal is Prime
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On Finitely Generated Module Whose First Nonzero Fitting Ideal is Prime Somayeh Hadjirezaei1 Received: 10 September 2019 / Accepted: 30 December 2019 © Iranian Mathematical Society 2020
Abstract 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. The main result of this paper is to characterize modules whose first nonzero Fitting ideals are prime ideals, in some cases. As a consequence, it is shown that if M is an Artinian R-module and I(M) = q is a prime ideal of R which contains a nonzero divisor, then M ∼ = R/q ⊕ P, for some submodule P of M. Keywords Fitting ideals · Torsion submodule · Regular element · Prime ideal Mathematics Subject Classification 13C05 · 13C12
1 Introduction Throughout this paper, R denotes a commutative ring with identity and all modules are unital. Original work by Fitting [5] showed how we can associate with each finitely generated R-module a unique sequence of ideals, which are known as Fitting ideals. The aim of this paper is to undertake an investigation of Fitting ideals and their relation with module structure, and to construct a module whose first nonzero Fitting ideal is a prime ideal. Buchsbaum and Eisenbud have shown in [1] that for finitely generated R-module M, I(M) = R if and only if M is a projective of constant rank module. A lemma of Lipman asserts that if R is a local ring and M = R m /K and I(M) is the (m − q)th Fitting ideal of M then I(M) is a regular principal ideal if and only if K is finitely generated free and M/ T(M) is free of rank m − q [10]. Fitting ideals are also used in mathematical physics. Einsiedler and Ward show how the dynamical properties of the system may be deduced from the Fitting ideals, and
Communicated by Mohammad Taghi Dibaei.
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Somayeh Hadjirezaei [email protected] Vali-e-Asr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran
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Bulletin of the Iranian Mathematical Society
they prove the entropy and expansiveness related with only the first Fitting ideal. This gives an easy computation instead of computing syzygy modules [2]. A partial list of important contributors to the theory of Fitting ideals includes the mathematicians Buchsbaum, Eisenbud, Fitting, Huneke, Katz, Lipman, Northcott, (for references for each author see [1,3,5,9,10,12]). Some recent works on Fitting ideals, due to author, characterize modules according to their First nonzero Fitting ideals [6–8]. We first introduce Fitting ideals and see how Fitting ideals arise by considering determinantal ideals of presentation matrices of the module and we describe some applications. We then study the behavior of Fitting ideals for certain module structures and investigate how useful Fitting ideals are in characterization of module.
2 Module Whose First Nonzero Fitting Ideal is a Prime Ideal First let us say definitions and statements. Let R be a commutative ring with identity and M be a finitely generated R-module. For a set {x1 , . . . , xn } of generators of M there is an exact sequence 0
Rn
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