Nil_＊-COHERENT RINGS

Abstract

Abstract. Let R be a ring and N il * (R) be the prime radical of R. In this paper, we say that a ring R is left N il * -coherent if N il * (R) is coherent as a left R-module. The concept is introduced as the generalization of left J-coherent rings and semiprime rings. Some properties of N il * -coherent rings are also studied in terms of N -injective modules and N -flat modules.

Nil$_{\ast}$-Noetherian rings

“…Recall from [9] that a ring R is called Nil * -coherent provided that any finitely generated ideal in Nil(R) is finitely presented. Similar to the classical case, Nil *coherent rings are not Nil * -Noetherian in general.…”

“…The famous generalization of the notion of Noetherian rings is that of coherent rings, i.e., rings in which any finitely generated ideal are finitely presented. For a further generalization, Xiang [9] introduced the notions of Nil * -coherent rings in terms of nil ideals in 2014. A ring R is said to be Nil * -coherent provided that any finitely generated nil ideal is finitely presented.…”

I-pure Submodules, I-FP-injective Modules and I-flat Modules

“…Recall that a left R-module M is F P -injective if and only if every R-homomorphism from a finitely generated submodule of a free left R-module F to M extends to a homomorphism of F to M [1, Proposition 2.6] . F P -injective modules and their generalizations have been studied by many authors, for example, see [6,7,10,11,12,13,14]. Following [11],…”

Nil$_{\ast}$-Artinian rings

“…Some algebraic researchers began to study rings by only consider their nil ideals. In 2014, Xiang [10] introduced the notions of Nil * -coherent rings in terms of nil ideals. A ring R is said to be Nil * -coherent provided that any finitely generated nil ideal is finitely presented.…”