Commutative rings and modules that are Nil*-coherent or special Nil*-coherent

Abstract

Recently, Xiang and Ouyang defined a (commutative unital) ring [Formula: see text] to be Nil[Formula: see text]-coherent if each finitely generated ideal of [Formula: see text] that is contained in Nil[Formula: see text] is a finitely presented [Formula: see text]-module. We define and study Nil[Formula: see text]-coherent modules and special Nil[Formula: see text]-coherent modules over any ring. These properties are characterized and their basic properties are established. Any coherent ring is a special Nil[Formula: see text]-coherent ring and any special Nil[Formula: see text]-coherent ring is a Nil[Formula: see text]-coherent ring, but neither of these statements has a valid converse. Any reduced ring is a special Nil[Formula: see text]-coherent ring (regardless of whether it is coherent). Several examples of Nil[Formula: see text]-coherent rings that are not special Nil[Formula: see text]-coherent rings are obtained as byproducts of our study of the transfer of the Nil[Formula: see text]-coherent and the special Nil[Formula: see text]-coherent properties to trivial ring extensions and amalgamated algebras.

Relative coherent modules

“…As an application of the previous results established in this section, we get the following result on the coherence of the amalgamated algebra alon an ideal which is proved differently in [1].…”

“…Proof. (1) Let N be an (n − 1)-presented module of X , then S ⊗ N is an (n − 1)-presented module of S⊗ X (since S is flat). Then, S ⊗ N is n-presented, so is N since S is faithfully flat.…”

Nil$_{\ast}$-Artinian rings

“…A ring R is said to be Nil * -coherent provided that any finitely generated nil ideal is finitely presented. Later in 2017, Ismaili et al [8] studied the Nil * -coherent properties via idealization and amalgamated algebras under several assumptions. Recently, Zhang [11] defined Nil * -Noetherian rings to be rings in which every nil ideal is finitely generated.…”

Nil$_{\ast}$-Noetherian rings

“…A ring R is said to be Nil * -coherent provided that any finitely generated nil ideal is finitely presented. Later in 2017, Alaoui Ismaili et al [1] studied the Nil * -coherent properties via idealization and amalgamated algebras under several assumptions.…”