Finiteness properties of generalized local cohomology modules for minimax modules

Let R be a commutative Noetherian ring, I an ideal of R, M be a finitely generated R-module and t be a non-negative integer. In this paper, we introduce the concept of I, M-minimax R-modules. We show that Hom R (R∕I, H I (M, N)∕K) is I,M-minimax, for all I,M-minimax submodules K of H I (M, N), whenever N and H I (M),


Introduction
Let R be a commutative Noetherian ring, I an ideal of R, and M a finitely generated R-module. An important problem in commutative algebra is determining when the set of associated primes of the i-th local cohomology module is finite. In Huneke, (1992) raised the following question: If M is a finitely generated R-module, then the set of associated primes of H i I (M) is finite for all ideals I of R and all i ≥ 0. Singh (2000) and Katzman (2002) have given counterexamples to this conjecture. However, it is known that this conjecture is true in many situations; see Brodmann and Lashgari Faghani (2000), Brodmann, Rotthous, and Sharp (2000), Hellus (2001), Marley (2001). In particular, Brodmann and Lashgari Faghani (2000) have shown that, Ass R (H t I (M)∕K) is a finite set for any finitely generated submodule K of H t I (M), whenever the local cohomology modules H 0 I (M), H 1 I (M), ⋯ , H t−1 I (M) are finitely generated. Next, Bahmanpour and Naghipour (2008) showed that, Hom R (R∕I, H t I (M)∕K) is finitely generated for any minimax submodule K of H t I (M), whenever the local cohomology modules H 0 I (M), H 1 I (M), ⋯ , H t−1 I (M) are minimax. After this Azami, Naghipour, and Vakili (2008) proved that, Hom R (R∕I, H t I (N)∕K) is I-minimax for any I-minimax submodule K of H t I (N), whenever N is an Iminimax R-module and the local cohomology modules H 0 The main result of this note is a generalization of above theorems for generalized local cohomology modules.
Recall that an R-module N is said to have finite Goldie dimension if N dose not contain an infinite direct sum of non-zero submodules, or equivalently the injective hall E(N) of N decomposes as a finite direct sum of indecomposable submodules. Also, an R-module N is said to have finite I-relative Throughout this paper, R will always be a commutative Noetherian ring with non-zero identity, I an ideal of R, M will be a finitely generated R-module and N an R-module. The i-th generalized local cohomology module with respect to I is defined by We refer the reader to Brodmann and Sharp (1998), Herzog (1974), Suzuki (1978), Yassemi, Khatami, and Sharif (2002), Payrovi, Babaei, and Khalili-Gorji (2015), Saremi and Mafi (2013) for the basic properties of local cohomology and generalized local cohomology.

I, M-minimax modules
For an R-module N the Goldie dimension is defined as the cardinal of the set of indecomposable submodule of E(N) which appear in a decomposition of E(N) in to a direct sum of indecomposable submodules. We shall use G dimN to denote the Goldie dimension of N. Let 0 ( , N) denote the 0-th Bass number of N with respect to prime ideal of R. It is well known that 0 ( , N) > 0 if and only if ∈ Ass R N and it is clear that Also, the I-relative Goldie dimension of N is defined as The I-relative Goldie dimension of an R-module has been studied in Divaani-Aazar and Esmkhani (2005) and in Lemma 2.6 it is shown that G dim I (N) = G dimH 0 I (N). Having this in mind, we introduce the following generalization of the notion of I-relative Goldie dimension.
Definition 2.1 Let I be an ideal of R and M be a finitely generated R-module. We denote by G dim I, M N the I, M-relative Goldie dimension of N and we define I, M-relative Goldie dimension of N as The class of I-minimax modules is defined in Azami et al. (2008) and an R-module N is said to be minimax with respect to I or I-minimax if I-relative Goldie dimension of any quotient module of N is finite. This motivates the following definition.  R (M, N)). N)). Therefore, we may assume that N is I-torsion free. Let E be an injective envelope of N and put N 1 = E∕N. Then

Finiteness of associated primes
It will be shown in this section that the subject of the previous section can be use to prove a finiteness result about generalized local cohomology modules. In fact we will generalize the main results of Brodmann and Lashgari Faghani (2000) and Azami et al. (2008). Throughout this section I is an ideal of R and M is a finitely generated R-module.