Ece Yetkin Celikel

In this paper, we introduce and study quasi-sdf-absorbing ideals, a generalization of sdf-absorbing ideals. We investigate the stability of this property under various constructions, including localizations, surjective images, Nagata idealizations, and amalgamations. We establish conditions under which the radical of such ideals is prime and discuss a specific class of rings where quasi-sdf-absorption implies the sdf-absorbing primary property. The study concludes with a classification of these ideals in [Formula: see text] and examples distinguishing them from related ideal classes.

Let R be a commutative ring with nonzero identity, and δ an expansion function of its ideals. In this paper, we introduce and study the concept of cube-difference factor absorbing δ(0)-ideals. A proper ideal I of R is called a cube-difference factor absorbing δ(0)-ideal (cdf-absorbing δ(0)-ideal, for short) if, for all a, b ∈ R, the condition a³ − b³ ∈ I implies that a − b ∈ I or a² + ab + b² ∈ δ(0). We establish various properties and characterizations of such ideals. We also examine the transfer of this property across several ring constructions, including polynomial rings, localizations, trivial ring extensions, and amalgamated algebras along an ideal. In particular, we characterize cdf-absorbing δ₀(0)-ideals of Z: among prime ideals via the third cyclotomic polynomial Φ3(x) = x² + x + 1 and its connection to Eisenstein primes, and among all ideals via a squarefreeness condition. We further extend the prime ideal characterization to arbitrary principal ideal domains.

In this paper, we introduce weakly classical 1-absorbing primary submodules of an unital module over a commutative ring with nonzero identity. Consider [ ] as a module over a ring [ ]. A proper submodule [ ] of [ ] is called weakly classical 1-absorbing primary submodule, if for each nonunit [ ] and [ ] with [ ], it follows that either [ ] or [ ] for some [ ]. Various properties and characterizations of weakly classical 1-absorbing primary submodules are explored.

Institute of Mathematics of the Czech Academy of Sciences

Let [ ] be a commutative ring with identity. A proper ideal [ ] of a ring [ ] is called a square-difference factor absorbing primary ideal of [ ] if for [ ], whenever [ ], then [ ] or [ ]. Several characterizations and properties of this class of ideals are presented. Various examples are provided to illustrate the obtained results and demonstrate the applicability of our findings. Furthermore, the properties of this class of ideals are investigated in extensions of rings.

Let R be a commutative ring with identity and M an unitary R- module. Recently, in [5], Anderson, Badawi and Coykendalla defined a proper ideal I of R to be a square-difference factor absorbing ideal (sdf-absorbing ideal) of R if whenever a ² − b ² ∈ I for 0 ≠ a, b ∈ R , then a + b ∈ I or a − b ∈ I . Generally, this article is devoted to introduce and study square-difference factor absorbing submodules. A proper submodule N of M is called square-difference factor absorbing (sdf-absorbing) in M if whenever m ∈ M and a, b ∈ R\Ann _R ( m ) such that ( a ² − b ² ) m ∈ N , then ( a + b ) m ∈ N or ( a − b ) m ∈ N . Many properties, examples and characterizations of sdf-absorbing submodules are introduced, especially in multiplication modules. Comparing this new class of submodules with classical prime submodules, we present new characterizations for von-Neumann regular modules in terms of sdf-absorbing submodules. Further characterizations of some special modules in which every nonzero proper submodule is sdf-absorbing are investigated. Finally, the sdf-absorbing submodules in amalgamated modules are studied.

Institute of Mathematics of the Czech Academy of Sciences