Extension functors of cominimax modules. Issue 2 (1st February 2017) - Hassanzadeh-Lelekaami, D.; Roshan-Shekalgourabi, H.
ABSTRACT: Let R be a commutative Noetherian ring with identity and I be an ideal of R . Also, let M and N be two nonzero R -modules. We prove that the R -module E x t R i ( N, M ) is I -cominimax for all i ≥0, whenever M is I -cominimax and N is finitely generated with dim N  ≤ 2. Also, it is shown that the R -module E x t R i ( N, M ) is I -cominimax for all i  ≥ 0, whenever N is finitely generated and M is I -cominimax with dim M  ≤ 1. As an immediate consequence, we obtain that if M is a nonzero minimax R -module such that dim H I i ( M ) ≤ 1, then for each finitely generated R -module N, E x t R j ( N, H I i ( M ) ) is I -cominimax for all i  ≥ 0 and j  ≥ 0. Moreover, we prove that if R is local, M is I -cominimax and N is finitely generated, then the R -module E x t R i ( N, M ) is I -weakly cofinite for all i  ≥ 0, when one of the following statements holds: (i) dim N  = 3 or (ii) dim M  ≤ 2.