Trivial extensions defined by 2-absorbing-like conditions

Abstract

Let [Formula: see text] be a commutative ring with [Formula: see text] The notions of 2-absorbing ideal and 2-absorbing primary ideal are introduced by Ayman Badawi as generalizations of prime ideal and primary ideal, respectively. A proper ideal [Formula: see text] of [Formula: see text] is called a 2-absorbing ideal of [Formula: see text] (respectively, 2-absorbing primary ideal) if whenever [Formula: see text] with [Formula: see text] then [Formula: see text] or [Formula: see text] or [Formula: see text] (respectively, [Formula: see text] or [Formula: see text] or [Formula: see text]). In this paper, we investigate the transfer of 2-absorbing-like properties to trivial ring extensions.

Ding projective and Ding injective modules over trivial ring extensions

“…This ring is called the trivial extension of the ring R by the bimodule M, and denoted by R ⋉ M. When R is a commutative ring, Nagata also called this construction an idealization in [21]. The notion of trivial extension of a ring by a bimodule is an important extension of rings and has played a crucial role in ring theory and homological algebra [2,8,15,19,21,22,23]. Fossum, Griffith and Reiten studied the categorical aspect and homological properties of trivial ring extensions [8].…”

On $n$-absorbing prime ideals of commutative rings

“…The concept of n-absorbing ideals is a generalization of the concept of prime ideals (note that a prime ideal of R is a 1-absorbing ideal of R). For more details on n-absorbing ideals, we refer the reader to [11][12][13]. We investigate rings in which every n-absorbing ideal of R is a prime ideal, where n ≥ 2 is an integer, called n-AB rings.…”