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In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

A proper ideal I of a commutative ring R is said to be a strongly quasi-primary ideal if, whenever a , b ∈ R with a b ∈ I , then a 2 ∈ I or b ∈ I (see [S. Koc, U. Tekir and G. Ulucak, On strongly quasi primary ideals, Bull. Korean Math. Soc. 56(3) (2019) 729–743]). This paper studies the class of strongly quasi-primary ideals with a radical equal to the nil-radical of R , called the class of quasi- n -ideals. Among other results, this new class of ideals is used to characterize when the nil-radical of R is a maximal or a minimal ideal of R. Many examples are given to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

Two presentations of quantum affine superalgebras were introduced by Yamane in [On defining relations of affine Lie superalgebras and affine quantized universal enveloping superalgebras, Publ. Res. Inst. Math. Sci. 35 (1999) 321–390], which were called Drinfeld–Jimbo realization and Drinfeld realization. Drinfeld realization contains infinite sequences of generators and relations. In this paper, we consider the Drinfeld realization of quantum affine superalgebra q ( ̂ m | n) associated to type m | n and define a simple algebra 0 ( ̂ m | n) generated by only a finite part of these sequences of quantum affine superalgebra q ( ̂ m | n). We show that the algebra 0 ( ̂ m | n) is isomorphic to the quantum affine superalgebra q ( ̂ m | n). Using the above isomorphism, we prove there exists an isomorphism between the two realizations. [ABSTRACT FROM AUTHOR]

In [D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467–479], Rees gave a characterization for the normal joint reduction number zero of two 픪-primary ideals in an analytically unramified Cohen–Macaulay local ring of dimension two. Rees’ result is a generalization of Zariski’s product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees’ theorem for the ordinary powers of 픪-primary ideals I and J in a Cohen–Macaulay local ring of dimension two. Following Rees’ approach, we define the modified Koszul homology modules Mr,s1(ak,bk) for a joint reduction (a,b) of I and J. Under the additional assumption that the associated graded rings of I and J have positive depth, we obtain a characterization of the joint reduction number zero of I and J in terms of the vanishing of the module M0,01(a,b), as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of Mr,s1(a,b) for any r,s ≥ 0. This gives a characterization for a vector (r,s) to be in the joint reduction lattice of I and J. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector (r,s) to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in [Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra 218(5) (2014) 904–918] for ordinary powers of ideals. [ABSTRACT FROM AUTHOR]

The well-known notion of an extending module is closely linked to that of a Baer module. A right R -module M is called extending if every submodule of M is essential in a direct summand. On the other hand, a right R -module M is called Baer if for all N ≤ M , l S (N) ≤ ⊕ S S where S = End R (M). In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc.  21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: "a module M is -nonsingular extending if and only if M is -cononsingular Baer". M R is called -nonsingular if ∀ φ ∈ S such that Ker φ ≤ e M , φ = 0. Moreover, M R is called -cononsingular if for any N ≤ M with φ N ≠ 0 for all 0 ≠ φ ∈ S , implies N ≤ e M. In view of this result, every Baer module which happens to be -cononsingular will automatically become an extending module. In this paper, our main focus is the study of -cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of -cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of -cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman's result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a condition on a p-subgroup of a finite group that is more general than the concept of ICΦ-subgroups, and obtain a sufficient and necessary condition for p-nilpotence of a finite group. We not only simplify the proofs of some main theorems of Gao and Li [Acta Math. Hungar. 163(1) (2021) 29–36], Kaspczyk [Acta Math. Hungar. 165(2) (2021) 355–359], and Kaspczyk [J. Algebra Appl. 23(1) (2024) 2450016], but also generalize these results. [ABSTRACT FROM AUTHOR]

The aim of this work is to prove a technical result that had been stated by Boyallian et al. [Classification of finite irreducible modules over the Lie conformal superalgebra C K 6 , Comm. Math. Phys. 317 (2013) 503–546], on the degree of singular vectors of finite Verma modules over the exceptional Lie superalgebra E (1 , 6) that is isomorphic to the annihilation superalgebra associated with the conformal superalgebra C K 6 . [ABSTRACT FROM AUTHOR]