Communications in Algebra

Staic, Mihai D.

doi: 10.1080/00927872.2023.2171426pmid: N/A

In this paper we introduce a generalization of the exterior algebra of a vector space V. We study some of its properties and give explicit computations for the case . In particular we show the existence and uniqueness of a determinant-like function with the property that if there exists such that . We also introduce the notion of GSC-operads and show how fits in that context.

Nikseresht, Azadeh

doi: 10.1080/00927872.2023.2172175pmid: N/A

Abstract A henselian valued field is called defectless (resp. tame) if each of its finite extensions is defectless (resp. tame). For a henselian field K in [3], a question was posed: “If every simple algebraic extension of K is defectless, then is it true that K is defectless?” An example showed that the answer is “no” in general. In this paper, we show that the analogue of this question for tame fields has an affirmative answer. Indeed it is proved that if every simple algebraic extension of a henselian field K is tame, then it is a tame field. Moreover, for an algebraic element θ over a tame field K, it is known that all those elements appearing in a saturated distinguished chain for θ stay inside of . This rises the problem of studying conditions under which the next element to θ in a chain does not necessarily stay inside of . Here we will give a sufficient condition for when there is no algebraic element such that is a distinguished pair.

Finnegan, Conor; Higgs, Russell J.

doi: 10.1080/00927872.2023.2172176pmid: N/A

Abstract Let ω be a 2-cocycle of a metacyclic p-group G representing a non-trivial element of the Schur multiplier Then the number of ω-regular conjugacy classes of G, the subgroup consisting of the ω-regular elements in the center of G, the degree of each irreducible ω-character of G and a representation group H of G with M(H) trivial are all determined. Finally, for ω constructed from H, the projective character table of G corresponding to ω is found in the case that G is of positive type. Communicated by Mark Lewis

Oman, Greg; Werner, Nicholas J.

doi: 10.1080/00927872.2023.2172177pmid: N/A

Abstract Let R be an associative ring. Then R is said to be coverable provided R is the union of its proper subrings (which we do not require to be unital even if R is so). One verifies easily that R is coverable if and only if R is not generated as a ring by a single element. In case R can be expressed as the union of a finite number of proper subrings, the least such number is called the covering number of R. Covering numbers of rings have been studied in a series of recent papers. The purpose of this note is to study rings which can be covered by a countable collection of subrings.

Dong, Yanyan; Shi, Fu-Gui

doi: 10.1080/00927872.2023.2172178pmid: N/A

Abstract The aim of this paper is to introduce convex structures on MV-algebras such that the MV-operations are convexity preserving or weak convexity preserving. Therefore, we propose the concepts of paraconvex MV-algebras and weak convex MV-algebras. We give some characterizations of weak convex MV-algebras. Further, we show that the standard MV-algebra endowed with its interval convexity is a weak convex MV-algebra. In particular, a finite MV-chain endowed with a non-trivial convex structure is a weak convex MV-algebra iff the convex structure is precisely its interval convexity. Moreover, the direct product of finite weak convex MV-algebras is still a weak convex MV-algebra. Based on this, we further get that each finite MV-algebra endowed with its interval convexity is a weak convex MV-algebra. By using ideals, we introduce the ideal convexity on an MV-algebra which turns it to be a paraconvex MV-algebra. Finally, we discuss the separation axioms on weak convex MV-algebras. Communicated by Ángel del Río Mateos

Baghdari, Samaneh; Öinert, Johan

doi: 10.1080/00927872.2023.2172179pmid: N/A

In this article, we provide a complete characterization of abelian group rings which are Köthe rings. We also provide characterizations of (possibly non-abelian) group rings over division rings which are Köthe rings, both in characteristic zero and in prime characteristic, and prove a Maschke type result for pure semisimplicity of group rings. Furthermore, we illustrate our results by several examples. Communicated by Eric Jespers

Güloğlu, İsma i°l Ş.; Ercan, Gülin

doi: 10.1080/00927872.2023.2172180pmid: N/A

Abstract Let A be a group acting by automorphisms on the group G. The commuting graph of A-orbits of this action is the simple graph with vertex set , the set of all A-orbits on , where two distinct vertices xA and yA are joined by an edge if and only if there exist and such that . The present paper characterizes the groups G for which is an -graph, that is, a connected graph which contains at most one vertex whose degree is not less than three.

Asaad, M.; Ramadan, M.

doi: 10.1080/00927872.2023.2173128pmid: N/A

Abstract Let G be a finite group, H a subgroup of G and p a prime number. We say that H is weakly s p -permutable in G if G has a subnormal subgroup K such that G = HK, and is a -number, where HsG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. In this paper, we investigate the structure of a group G under the assumption that certain subgroups of G are weakly s p -permutable in G. Some recent results are extended and generalized. Communicated by Alexander Olshanskii

Çetin, Miraç; Koşan, M. Tamer; Žemlička, Jan

doi: 10.1080/00927872.2023.2173762pmid: N/A

Abstract This paper describes properties of three certain classes of rings determined by conditions on idempotents and units, namely, the condition that any two generators of each principal right ideal are associated (UG rings), the condition that every principal right ideal is generated by a sum of a unit and an idempotent (Pr ), and the condition xy = 0 implies xsy = 0 for a sum of idempotent and unit s and any elements x, y of a ring (idun-semicommutative rings). It is proved that the class of all UG rings contains every local as well as every von Neumann regular ring, and the condition Pr is satisfied by both semiperfect and regular rings. Both local and abelian regular rings are proved to be necessarily idun-semicommutative. For all three classes are presented some closure properties and illustrating examples.

Gyoda, Yasuaki

doi: 10.1080/00927872.2023.2173763pmid: N/A

Abstract In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the -tilting theory of cluster-tilted algebras of finite representation type using the correspondence between clusters and support -tilting modules.