In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring

Rings in which each finitely generated right ideal is automorphism-invariant (rightfa-rings) are shown to be isomorphic to a formal matrix ring. Among other results it is also shown that (i) if R is a right nonsingular ring and n>1 is an integer, then R is a right self injective regular ring if and only if the matrix ring 𝕄n(R) is a right fa-ring, if and only if 𝕄n(R) is a right automorphism-invariant

In 2004, Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a ℤ-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that cannot be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.

We study canonical bases of a subalgebra A=𝕂[[f1,…,fs]]⊆𝕂[[x1,…,xn]] over a field 𝕂, and we associate with A a fan called the canonical fan of A. This generalizes the notion of the standard fan of an ideal.

In this paper, we mainly investigate the class-preserving Coleman automorphisms of finite groups whose Sylow 2-subgroups are semidihedral. We prove that if G is a finite solvable group with semidihedral Sylow 2-subgroups, then Outc(G)∩OutCol(G) is a 2′-group and therefore G satisfies the normalizer property. As some applications of this result, we also investigate the normalizer property of the following

We study some properties of formal local cohomology modules ℱ𝔞i(M) in Serre subcategories. As a consequence, we obtain some results on the minimax modules. We also describe the sets CosR(ℱ𝔞i(M)) and CoassR(ℱ𝔞i(M)).

A first aim of this paper is to give sufficient conditions on left non-degenerate bijective set-theoretic solutions of the Yang–Baxter equation so that they are non-degenerate. In particular, we extend the results on involutive solutions obtained by Rump in A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation, Adv. Math. 193 (2005) 40–55, https://doi.org/10

The Bochner–Schoenberg–Eberlein (BSE)-property on commutative Banach algebras is a property related to multiplier algebras of Banach algebras. In this paper, we answer the problem (12) raised by Javanshiri and Nemati in [Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜, J. Algebra Appl.17(9) (2018) 1850169-1–1850169-21]. In this paper, under certain conditions, we show that the

In this paper, we prove that if R is a local ring of dimension d,d≥2 and 1d!∈R then the group Umd+1(R[X])Ed+1(R[X]) has no k-torsion, provided kR=R. We also prove that if R is a regular ring of dimension d,d≥2 and 1d!∈R such that Ed+1(R) acts transitively on Umd+1(R) then Ed+1(R[X]) acts transitively on Umd+1(R[X]).

In this paper, we study the behavior of endomorphism rings of indecomposable (uniform) quasi-injective modules. A very natural question here is, for a morphism f:A→B, with A,B indecomposable (uniform) quasi-injective right R-modules, and g:E(A)→E(B) an extension of f where E(−) denotes the injective hull, what is the relation between kernels of f and g, their monogeny classes and their upper parts

We introduce the notion of semi-poor modules and consider the possibility that all modules are either injective or semi-poor. This notion gives a generalization of poor modules that have minimal injectivity domain. A module M is called semi-poor if whenever it is N-injective and N≠0, then the module N has nonzero socle. In this paper the properties of semi-poor modules are investigated and are used

We construct an example of an IC ring where perspectivity is transitive, but not all isomorphic idempotents are perspective. We also develop new criteria for checking perspectivity of idempotents in rings.

Let T=A0UB be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. We give some computing formulas of homological dimensions of special T-modules. As an application, we describe the structures of n-tilting left T-modules.

Given a finite-dimensional algebra Λ and A≥1, we construct a new algebra Λ̃A, called the stretched algebra, and relate the homological properties of Λ and Λ̃A. We investigate Hochschild cohomology and the finiteness condition (Fg), and use stratifying ideals to show that Λ has (Fg) if and only if Λ̃A has (Fg). We also consider projective resolutions and apply our results in the case where Λ is a d-Koszul

In this paper, we introduce and study the notion of quasi-multipliers on a semi-topological semigroup S. The set of all quasi-multipliers on S is denoted by 𝔔𝔐(S). First, we study the problem of extension of quasi-multipliers on topological semigroups to its Stone–Čech compactification. Indeed, we prove if S is a topological semigroup such that S×S is pseudocompact, then 𝔔𝔐(S) can be regarded as

Let R be a commutative Noetherian ring. For a finitely generated R-module M, Northcott introduced the reducibility index of M, which is the number of submodules appearing in an irredundant irreducible decomposition of the submodule 0 in M. On the other hand, for an Artinian R-module A, Macdonald proved that the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation

Let G be a finite group, p a prime, P a Sylow p-subgroup of G and d a power of p such that 1

Let R be a unital ring, let E be a directed graph and recall that the Leavitt path algebra LR(E) carries a natural ℤ-gradation. We show that LR(E) is strongly ℤ-graded if and only if E is row-finite, has no sink, and satisfies Condition (Y). Our result generalizes a recent result by Clark, Hazrat and Rigby, and the proof is short and self-contained.

In this paper, we investigate the properties of well-rounded twists of a given ideal lattice of an imaginary quadratic field K. We show that every ideal lattice I of K has at least one well-rounded twist lattice. Moreover, we provide an explicit algorithm to compute all well-rounded twists of I.

Rota–Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota–Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum

There is a special local ring E of order 4, without identity for the multiplication, defined by E=〈a,b|2a=2b=0,a2=a,b2=b,ab=a,ba=b〉. We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over E, and Type IV codes, that is quasi self-dual codes whose all codewords have even

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as E=〈a,b|2a=2b=0,a2=a,b2=b,ab=a,ba=b〉. We study a recursive construction of self-orthogonal codes over E. We classify, up to permutation equivalence, self-orthogonal codes of length n and size 2n (called here quasi self-dual codes or QSD) up to the length n=12. In particular, we classify

In this paper, we introduce a class of commutative rings which is a generalization of ZD-rings and rings with Noetherian spectrum. A ring R is called stronglyJ-Noetherian whenever the ring Rr is J-Noetherian for every non-nilpotent r∈R. We give some characterizations for strongly J-Noetherian rings and, among the other results, we show that if R[[x]] is strongly J-Noetherian, then R has Noetherian

We give an example to show that, for nonunital rings R, the direct sum aR⊕bR=(a+b)R with a+b regular has no in general right-left symmetry. It is then proved that the right-left symmetry actually holds in a left and right faithful ring.

Let X be a finite partially ordered set, R a commutative domain and σ an endomorphism of R. We define the skew incidence ring I(X,R,σ) and we prove that each Jordan isomorphism of I(X,R,σ) onto an associative ring S is a near-sum of a homomorphism and an anti-homomorphism.

In this paper, we introduced the concept of crossed modules for Hom–Lie antialgebras. It is proved that the category of crossed modules for Hom–Lie antialgebras and the category of Cat1-Hom–Lie antialgebras are equivalent to each other. The relationship between the crossed modules extension of Hom–Lie antialgebras and the third cohomology group is investigated.

In this paper, we introduce ∗-almost independent rings of Krull type (∗-almost IRKTs) and ∗-almost generalized Krull domains (∗-almost GKDs) in the general theory of almost factoriality, neither of which need be integrally closed. This fills a gap left in [D. D. Anderson and M. Zafrullah, On∗-Semi-Homogeneous Integral Domains, Advances in Commutative Algebra (Springer, Singapore, 2019)]. We characterize

We characterize bijective linear maps on Mn(ℂ) that preserve the square roots of an idempotent matrix (of any rank). Every such map can be presented as a direct sum of a map preserving involutions and a map preserving square-zero matrices. Next, we consider bijective linear maps that preserve the square roots of a rank-one nilpotent matrix. These maps do not have standard forms when compared to similar

A few years ago the so-called oriented subgroup F→ of the Thompson group F was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of F it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, F→ provided the first example of a maximal subgroup of infinite index in F different

We classify, in terms of the structure of the finite group G, all group algebras KG for which all right ideals are right annihilators of principal left ideals. This means in the language of coding theory that we classify code-checkable group algebras KG which have been considered so far only for abelian groups G. Optimality of checkable codes and asymptotic results are discussed.

Kayal and Nezhmetdinov introduced in [Factoring groups efficiently, Int. Colloquium on Automata, Languages, and Programming (Springer, Berlin, Heidelberg, 2009), pp. 585–596] the concept of an irreducible conjugacy class and defined, for any finite non-abelian group, a simple graph on the set of all such classes. This graph is a key ingredient in their proposed algorithm for solving the group decomposition

In this paper, we mainly consider three classes of quantum B-algebras such as locally unital quantum B-algebras, two-sided quantum B-algebras and implication-commutative quantum B-algebras. We first prove that a quantum B-algebra can be embedded into a unital quantum B-algebra if and only if it is a locally unital quantum B-algebra. Next, we introduce two-sided (implication-commutative) quantum B-algebras

In this paper we introduce dual F-Baer modules and give a characterization of them. Let M be a module and F a fully invariant submodule of M. M is called dual F-Baer if for every family of endomorphisms {gα}α∈I of M, ∑α∈Igα(F) is a direct summand of M. We prove that M is dual F-Baer if and only if M=F⊕N for some submodule N of M with F dual Baer. We obtain a positive solution for the Schröder–Bernstein

We consider HNN extensions 〈B,t:t−1Ht=K〉, where H,K are normal subgroups of B and H∩K=1. We show that class-preserving automorphisms of those HNN extensions are all inner if B satisfies certain conditions. From Grossman’s result, it follows that outer automorphism groups of such conjugacy separable HNN extensions are residually finite.

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a Gröbner–Shirshov basis theory framework for modules over associative conformal algebras and apply this technique to Poisson algebras considered as conformal modules over appropriate associative conformal envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation

Let 𝒞 be small category and 𝒜 an arbitrary category. Consider the category 𝒞(𝒜) whose objects are functors from 𝒞 to 𝒜 and whose morphisms are natural transformations. Let ℬ be another category, and again, consider the category 𝒞(ℬ). Now, given a functor F:𝒜→ℬ we construct the induced functor F𝒞:𝒞(𝒜)→𝒞(ℬ). Assuming 𝒜 and ℬ to be abelian categories, it follows that the categories 𝒞(𝒜)

The norm N(G) of a group G is the intersection of the normalizers of all subgroups in G. In this paper, the norm is generalized by studying on Sylow subgroups and ℋ-subgroups in finite groups which is denoted by C(G) and A(G), respectively. It is proved that the generalized norms A(G) and C(G) are all equal to the hypercenter of G.

Let B be an one-point extension of a finite-dimensional k-algebra A by a simple A-module at a source point i. In this paper, we classify the τ-tilting modules over B. Moreover, it is shown that there are equations |τ−tiltB|=|τ−tiltA|+|τ−tiltA/〈ei〉|and|sτ−tiltB|=2|sτ−tiltA|+|sτ−tiltA/〈ei〉|. As a consequence, we can calculate the numbers of τ-tilting modules and support τ-tilting modules over linearly

A submodule W of V is summand absorbing, if x+y∈W implies x∈W,y∈W for any x,y∈V. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules.

The aim of this paper is to give a new characterization of cyclic modules over local rings, namely, we show that if N is a nonzero finitely generated module over a local (Noetherian) ring (R,𝔪) such that for some integer n≥1, the submodule 𝔪nN of N is cyclic, then every submodule of N is cyclic and the ring R/AnnR(N) is a DVR, whenever depthN>0. As a consequence, it follows that R/AnnR(N/Γ𝔪(N))

Let R be a commutative ring with identity and S a multiplicative subset of R. In this paper, the notion of S-pseudo-irreducible ideals of R as a generalization of pseudo-irreducible ideals is introduced and some related properties are investigated. Then, we consider the decomposition of proper ideals of a ring into a product of pairwise comaximal S-pseudo-irreducible ideals. Some other results are

We continue a very fruitful line of inquiry into the multiplicative ideal theory of an arbitrary Leavitt path algebra L. Specifically, we show that factorizations of an ideal in L into irredundant products or intersections of finitely many prime-power ideals are unique, provided that the ideals involved are powers of distinct prime ideals. We also characterize the completely irreducible ideals in L

In this paper, we describe absolute nilpotent and some idempotent elements of an n-dimensional evolution algebra corresponding to two permutations and we decompose such algebras to the direct sum of evolution algebras corresponding to cycles of the permutations.

In this paper, we address the question of finding the length of group algebras of finite abelian groups in the case when the characteristic of the ground field divides the order of the group. We evaluate the exact length for an arbitrary abelian p-group. For other groups we provide upper and lower bounds for the length of their group algebras.

In this paper, the commuting graphs associated with finite groups are discussed. As the main theme, the explicit formulas for the number of spanning trees of commuting graphs associated with some specific groups, such as the Suzuki simple groups, the projective special linear groups, and some certain AC-groups, are obtained.

We study the universal (associative) envelope of the Jordan triple system of all n×n(n≥2) matrices with the triple product {x,y,z}=xyz+zyx over a field of characteristic 0. We use the theory of non-commutative Gröbner–Shirshov bases to obtain the monomial basis and the center of the universal envelope. We also determine the decomposition of the universal envelope and show that there exist only five

In this paper, we study the pullbacks of LPVDs (respectively, t-LPVDs). We also study the homological properties of LPVDs (respectively, t-LPVDs). More specifically, it is proved that the only possible weak global dimensions (respectively, w-weak global dimensions) of an LPVD (respectively, a t-LPVD) are 0,1,2, and ∞.

Let R be a commutative ring with non-zero identity, S⊆R a multiplicatively closed subset of R and M a unital R-module. In this paper, we introduce and study the concept of S-1-absorbing prime submodules. A submodule N of M with (N:RM)∩S=∅ is said to be S-1-absorbing prime, if there exists an s∈S whenever abm∈N for some non-unit elements a,b∈R and m∈M, then sab∈(N:RM) or sm∈N. Some examples, characterizations

In this paper, we study pre-Lie analogues of Poisson-Nijenhuis structures and introduce 𝒪N-structures on bimodules over pre-Lie algebras. We show that an 𝒪N-structure gives rise to a hierarchy of pairwise compatible 𝒪-operators. We study solutions of the strong Maurer–Cartan equation on the twilled pre-Lie algebra associated to an 𝒪-operator, which gives rise to a pair of 𝒪N-structures which are

Let R be an associative ring equipped with an automorphism α and an α-derivation δ. Necessary and sufficient conditions are obtained for the skew inverse Laurent series ring R((x−1;α,δ)) to satisfy a certain ring property which is among being local, semilocal, semiperfect and semisimple, respectively. We also give a characterization of the cleanness of the ring R((x−1;α,δ)).

For n≥2 and for a ring R, the notation Pn(R) means that rn−r is nilpotent for all r∈R. In this paper, rings R for which Pn(R) holds are completely characterized for any integers n≥2. This answers a question which was raised in [T. Kosan, Y. Zhou, T. Yildirim, Rings with xn−x nilpotent, J. Algebra Appl.19(4) (2020) 2050065].

Let F be an infinite field and R be an algebraic F-algebra. It is known that if 𝒰(R), the unit group of R, satisfies a group identity, then R satisfies a polynomial identity. In this paper, using additive mappings, we generalize this result to some kinds of identities. Particularly, assume that ∗ is an involution of R. We show that if 𝒰(R) satisfies a ∗-group identity, then R satisfies a polynomial

Let A be a finitely generated associative algebra over a field F of characteristic ≠2 and let A(−)=(A,[a,b]=ab−ba) be its associated Lie algebra. In this paper, we investigate relations between the growth functions of A and the Lie algebra [A,A]. We prove that if A is generated by a finite collection of nilpotent elements, then the growth functions are asymptotically equivalent.

For a subset T of a ring R we denote by ℐ(T) the ideal of R generated by T. Given a higher commutator L of R, if R=ℐ(L) then R=L+L2? The question is motivated by the result that a ring R is equal to its subring generated by [R,R] if R is either a noncommutative simple ring (by Herstein) or a unital ring with 1∈[R,R] (by Eroǧlu). In this note, we study the question for the rings R satisfying the property

Based on the concept of a lattice preradical recently introduced in [T. Albu and M. Iosif, Lattice preradicals with applications to Grothendieck categories and torsion theories, J. Algebra444 (2015) 339–366], we present and investigate in this paper, the latticial counterparts of the concepts of prime and irreducible preradicals on the category Mod-R of all unital right R-modules over an associative

Let ℰ be an injectively resolving subcategory of left R-modules. We introduce and study ℰ-Gorenstein flat modules as a common generalization of some known modules such as Gorenstein flat modules (Enochs, Jenda and Torrecillas, 1993), Gorenstein AC-flat modules (Bravo, Estrada and Iacob, 2018). Then we define a resolution dimension relative to the ℰ-Gorensteinflat modules, investigate the properties

In this paper, we focus on discussing diagonal solutions and general solutions of second-order matrix polynomial equation of high degree in complex field. By characterizing some algebraic properties of the mentioned two types of the solutions, we present sufficient conditions that a general second-order matrix polynomial equation has diagonal solutions or general solutions. Analytic expressions of

The big from small construction was introduced by Kustin and Miller in [A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra85 (1983) 303–322] and can be used to construct resolutions of tightly double linked Gorenstein ideals. In this paper, we expand on the DG-algebra techniques introduced in [A. Kustin, Use DG methods to build a matrix factorization, preprint (2019)

Let G be a finite group and H a subgroup of G. We say that H is an ℋ-subgroup of G if NG(H)∩Hg≤H for all g∈G; H is called weakly ℋC-embedded in G if G has a normal subgroup T such that HG=HT and NT(H)∩Hg≤H for all g∈G, where HG is the normal closure of H in G. In this paper, we study the p-nilpotence of a group G in which every subgroup of order d of a Sylow p-subgroup P with 1

Let 𝒪𝒫ℰn be the monoid of all orientation-preserving and extensive full transformations on {1,2,…,n} ordered in the standard way. In this paper, we determine the minimum generating set and the minimum idempotent generating set of 𝒪𝒫ℰn, and so the rank and the idempotent rank of 𝒪𝒫ℰn are obtained. Moreover, we describe maximal subsemigroups and maximal idempotent generated subsemigroups of 𝒪𝒫ℰn