Yazd University Algebraic Structures and Their Applications 2382-9761 7 1 2020 02 01 On the associated primes of the generalized $d$-local cohomology modules 1 10 1615 10.22034/as.2020.1615 EN Zahra Rahimi-molaei Department of Mathematics, Imam Khomeini International University, Qazvin, Iran Shiroyeh Payrovi Department of Mathematics, Imam Khomeini International University, Qazvin, Iran. Sakineh Babaei Department of mathematics, Imam Khomeini International University, Qazvin, Iran 0000-0001-7039-2095 Journal Article 2019 04 27 The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary  generalized local cohomology  modules.  Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$  are  finitely generated  $R$-modules and $d, t$ are two integers. We prove that $\Ass H^t_d(M,N)=\bigcup_{I\in \Phi} \Ass H^t_I(M,N)$ whenever $H^i_d(M,N)=0$ for all  $i< t$ and $\Phi=\{I: I  \text{ is an ideal of}\  R \text{ with}\ \dim R/I\leq d \}$. In the second part of the paper, we give some information about  the non-vanishing of the generalized $d$-local cohomology modules. To be more precise, we prove that $H^i_d(M,R)\neq 0$ if and only if $i=n-d$ whenever  $R$ is a Gorenstein ring of dimension $n$ and $pd_R(M)<\infty$. This result leads to an example which shows that $\Ass H^{n-d}_d(M,R)$ is not necessarily a finite set. https://as.yazd.ac.ir/article_1615_a0870992cd28c1d8c56ed39c2806d814.pdf