Let [Formula: see text] be a commutative ring with nonzero identity. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of [Formula: see text]-ideals. A proper ideal [Formula: see text] of [Formula: see text] is said to be a [Formula: see text]-ideal if whenever nonunit elements [Formula: see text] and [Formula: see text], then [Formula: see text] or [Formula: see text] Some basic properties of [Formula: see text]-ideals are studied. For instance, we give a method to construct [Formula: see text]-ideals that are not [Formula: see text]-ideals. Among other things, it is shown that if [Formula: see text] admits a [Formula: see text]-ideal that is not an [Formula: see text]-ideal, then [Formula: see text] is a local ring. Also, we provide a necessary and sufficient condition in term of [Formula: see text]-ideals for a ring to be a total quotient ring and we determinate the [Formula: see text]-ideals of a chained ring. Finally, we give an idea about some [Formula: see text]-ideals of the localization of rings, the trivial ring extensions and the power series rings to construct nontrivial and original examples of [Formula: see text]-ideals.
Let [Formula: see text] be a commutative ring with nonzero identity. A proper ideal [Formula: see text] of [Formula: see text] is called a 1-absorbing prime ideal (respectively, 1-absorbing primary ideal) if whenever nonunit elements [Formula: see text] with [Formula: see text] then [Formula: see text] or [Formula: see text] (respectively, [Formula: see text] or [Formula: see text]). The purpose of this paper is to study the transfer of certain 1-absorbing-like properties to amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text] (denoted by [Formula: see text]), introduced and studied by D’Anna, Finocchiaro and Fontana. Our results provide new techniques for the construction of new original examples satisfying the above-mentioned properties.
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