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We correct [3], Proposition 3.2] by showing that the class of quasi-Noetherian Lie algebras s is not E-closed. We repair the resulting gap in Example 5.1 of that paper by proving that any simple-by-soluble Lie algebra is quasi-Noetherian. More generally, any Noetherian-by-quasi-Noetherian Lie algebra is quasi-Noetherian. We also prove that any quasi-Noetherian-by-soluble Lie algebra is quasi-Noetherian, and prove E-closure for analogues of the quasi-Noetherian property for modules over associative rings and modules over Lie algebras. Using a wreath product for Lie algebras we prove that the quasi-Noetherian property is not inherited by ideals. Indeed, the derived algebra of a quasi-Noetherian Lie algebra need not be quasi-Noetherian. An analogous example is constructed for quasi-Artinian Lie algebras, which also shows that the Artinian and quasi-Artinian properties are not inherited by ideals of finite codimension. [ABSTRACT FROM AUTHOR]

This article explores the automorphisms of a Chevalley group of type G2 over a commutative ring R. The authors demonstrate that every automorphism of this group, when R is generated by the invertible elements and the ideal 2R, can be expressed as a combination of ring and inner automorphisms. The paper offers a historical overview of the study of automorphisms of classical groups and Chevalley groups, as well as the methods employed in previous research. The authors introduce definitions and main theorems related to Chevalley groups and their automorphisms. The text focuses on automorphisms of the group Gad(G2, R) and their properties, defining ring automorphisms and inner automorphisms, and establishing standard automorphisms as compositions of these two types. The primary objective is to prove that any automorphism of the group Gad(G2, R) is standard. The text also covers definitions and theorems concerning the localization of rings and modules, isomorphisms of Chevalley groups over fields, and the characteristic subgroup Ead(G2, R) in Gad(G2, R). The main theorem's proof is outlined in several steps, including the embedding of the ring R into a product of its localizations and the mapping of elements under conjugation by an element of Gad(G2, S). Additionally, a lemma is proven that demonstrates the mapping of matrices under conjugation by an element of Gad(G2, S) [Extracted from the article]

Regarding the construction of strict and base radical classes for the variety of associative rings, we show that "subring" and "accessible subring" are two of a number of substructures that can be used in constructing a radical class. The substructures we focus on are set properties that are transitive across accessible subrings. These substructures allow us to look beyond those that are related by the binary operations of the ring itself, to ones that may only be related by containment. The lower radical type construction defined by these substructures include those of Yu-lee Lee, McDougall and Stewart. [ABSTRACT FROM AUTHOR]

The goal of this paper is to describe the group of strongly stable automorphisms for categories of free finitely generated non-associative algebras with unit for classical varieties of non-associative algebras, in particular for the varieties of commutative, Jordan, alternative and power associative algebras. The motivation of this research is tightly connected with some problems in Universal Algebraic Geometry. [ABSTRACT FROM AUTHOR]

Let R be an associative unital ring, and let f ∈ R [ x ] . We say that f is a left annihilating content (AC) polynomial if f = af1 for some a ∈ R and f 1 ∈ R [ x ] with l R [ x ] (f 1) = 0 . The ring R is called a left EM-ring if each f ∈ R [ x ] is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element a ∈ R and a finitely generated right ideal J of R with l R (J) = 0 and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring R [ G ] is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, R [ x ] , R [ x , x − 1 ] , and R [ [ x ] ] are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings). [ABSTRACT FROM AUTHOR]

Let G be a group with a metric invariant under left and right translations, and let 픻r be the ball of radius r around the identity. A (k,r)-metric approximate subgroup is a symmetric subset X of G such that the pairwise product set XX is covered by at most k translates of X픻r. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of X finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on X replacing finiteness. In particular, if G has bounded exponent, we show that any (k,r)-metric approximate subgroup is close to a (1,r′)-metric approximate subgroup for an appropriate r′. [ABSTRACT FROM AUTHOR]

Let R be an associative and 2-torsion-free ring with an identity. in this work, we will generaliz the results of differentially prime rings in [18] by applying the hypotheses in a differentially semiprime rings. In particular, we have proved that if R is a D-semiprime ring, then either R is a commutative ring or D is a semiprime ring. [ABSTRACT FROM AUTHOR]

In this study, the new types of mappings on the rings and near-rings are defined, which are named multiplicative homoleft multipliers, multiplicative homoright multipliers and multiplicative homo multipliers. We also present examples that show the existence of such mappings. Moreover, we study how these mappings effected on the construction of near-rings. [ABSTRACT FROM AUTHOR]

A right module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. We show the closures properties for certain high submodules by the QTAG-modules and vice versa. Important generalizations and certain related assertions of classical results in this direction are also established. [ABSTRACT FROM AUTHOR]

An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup. [ABSTRACT FROM AUTHOR]

A square is a concatenation of two identical words, and a word w is said to have a square yy if w can be written as xyy z for some words x and z. It is known that the ratio of the number of distinct squares in a word to its length is less than two, and any location of a word could begin with two distinct squares which are appearing in the word for the last time. A square whose first location starts with the last occurrence of two distinct squares is an FS-double square. We explore and identify the conditions under which a sequence of locations in a word starts with FS-double squares. We first find the structure of a word that begins with two consecutive FS-double squares and obtain its properties that enable us to extend the sequence of FS-double squares. It is proved that the length of the longest sequence of consecutive FS-double squares in a word of length n is at most nn. We show that the squares in the longest sequence of consecutive FS-double squares are conjugates. [ABSTRACT FROM AUTHOR]

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

This paper presents a study on Jordan maps over matrix rings with some functional equations related to additive maps on these rings. We first show that every Jordan left (right) centralizer over a matrix ring is a left (right) centralizer. Moreover, every two-sided centralizer over the matrix ring is of a particular form. Further, we prove that any additive map satisfying functional equations over matrix rings becomes a two-sided centralizer. Finally, we conclude our work with some results on the Jordan left *- centralizer over matrix rings and establish some results on functional equations that arise for the *-centralizer. [ABSTRACT FROM AUTHOR]

Let R be an associative Noetherian unital noncommutative ring R. We introduce the functor PΓP over the category of R-modules and use it to characterize P-semiprime. P-semisecond R-modules also characterized by the functor PΛP. We also show that the Greenless-May type Duality (GM) and Matlis Greenless-May Equality(MGM) hold over the full subcategory of R-Mod consisting of P-semiprime and P-semisecond modules. Finally, we generate a one-sided right ideal PΓP(R), which gives an equivalent formulation to solve Köthe conjecture positively or negatively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of weakly semiprime ideals and weakly n -systems in noncommutative rings. We establish the equivalence between an ideal P being a weakly semiprime ideal and R − P being a weakly n -system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide. [ABSTRACT FROM AUTHOR]

For two given idempotents p and q from an associative algebra A , in this paper, we offer a comprehensive classification of algebras spanned by the idempotents p and q. This classification is based on the condition that p and q are not tightly coupled and satisfy (p q) m − 1 = (p q) m but (p q) m − 2 p ≠ (p q) m − 1 p for some m (≥ 2) ∈ N. Subsequently, we categorize all the group invertible elements and establish an upper bound for the Drazin index of any elements in these algebras spanned by p , q. Moreover, we formulate a new representation for the Drazin inverse of α p + q under two different assumptions, (p q) m − 1 = (p q) m and λ (p q) m − 1 = (p q) m , where α is a non-zero and λ is a non-unit real or complex number. [ABSTRACT FROM AUTHOR]

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants p , n , s , and t , where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of p 6 and p 7 that hold substantial importance in the field of coding theory. [ABSTRACT FROM AUTHOR]

The paper contains the classification of non-commutative torsion-free associative rings of rank two. Furthermore, torsion-free abelian groups of rank two supporting an associative, but not commutative ring structure are classified. [ABSTRACT FROM AUTHOR]

This paper deals with additive groups of rings in which all subrings are ideals. It is shown that if an abelian group supports only rings with this property, then all of them are commutative. This result is obtained for associative as well as not necessarily associative rings. Some previously known far from obvious results related to the mentioned groups are generalized and complemented. In particular, the classification of torsion-free abelian groups on which every not necessarily associative ring has the property that all its subrings are ideals is provided up to the structure of nil-groups. [ABSTRACT FROM AUTHOR]

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k ≥ 2 , an irreducible module of dimension 3k is also constructed. [ABSTRACT FROM AUTHOR]

Let R be an associative unital ring and α an endomorphism of R. In this article, we aim to investigate rings that satisfy Property (a.c.) and explore some related rings. Moreover, we examine many polynomial extensions of R satisfying Property (a.c.). In particular, α-skew quasi-Armendariz rings and α-(sps) quasi-Armendariz rings. We prove that this class of rings has always Property (a.c.) on the left but not necessarily on the right. Furthermore, we investigate the transmission of Property (a.c) from a ring R to R[x, α] and R[[x, α]]. [ABSTRACT FROM AUTHOR]

On the ring M n (C) of all n × n complex matrices A, the conjugate transpose involution A ↦ A * has the "properness" property A * A = 0 ⇒ A = 0 , as required for the existence of the partial order known as the " * -order" on M n (C) . More generally, related questions are asked and answered about the ring M n (R) of all n × n matrices over an arbitrary associative ring R with any given involution R → R (as a generalization of the conjugacy map C → C ), yielding a corresponding " * -transposition" involution on M n (R) . A criterion is found for this involution of M n (R) to be proper, so that M n (R) has a corresponding * -order. The special case R = Z k is also considered. [ABSTRACT FROM AUTHOR]

BiHom-associative algebras are generalized associative algebras with two multiplicative linear maps. In this paper, we give the Hochschild homology and cyclic homology structure of BiHom-associative algebras. Then, generalize the dual bimodule action to define the cyclic cohomology. Finally, we introduce the crossed modules of BiHom-associative algebras and show that the Hochschild cohomology of BiHom-associative algebra classifies crossed modules. [ABSTRACT FROM AUTHOR]

The representation of elements of free non-associative algebras as a set of multidimensional tables of coefficients is defined. An operation for finding partial derivatives for elements of free non-associative algebras in the same form is considered. Using this representation, a criterion of primitivity for elements of lengths 2 and 3 in terms of matrix ranks, as well as a primitivity test for elements of arbitrary length, is derived. This test makes it possible to estimate the number of primitive elements in free non-associative algebras with two generators over a finite field. The proposed representation allows us to optimize algorithms for symbolic computations with primitive elements. Using these algorithms, we find the number of primitive elements of length 4 in a free non-associative algebra of rank 2 over a finite field. [ABSTRACT FROM AUTHOR]

An associative ring R, not necessarily commutative and not necessarily with identity, is called a BN ring if every element of R is either idempotent or nilpotent. If the index of nilpotency of the elements is bounded by k, then we call a BN ring R a BN(k) ring. A BN ring is proper if it is neither Boolean nor nil. We show that any proper BN(k) ring is a BZS, that is a BN(2), ring. [ABSTRACT FROM AUTHOR]

It is well known that any Lie algebra can be embedded into an associative algebra. We prove that any metabelian Lie algebra can be embedded into an algebra in the subvariety of perm algebras, i.e. associative algebras with the identity a b c − a c b = 0. In addition, a technical method to construct the universal enveloping perm algebra for a metabelian Lie algebra is given. [ABSTRACT FROM AUTHOR]

In this paper, we define a G -graded Hopf module associative pseudoalgebra A for any grading group G and Hopf algebra H , and construct a smash product A # H of A with H , which is also a G -graded associative pseudoalgebra. Then we establish a Mortia context between A # H and A H , the fixed sub-pseudoalgebra under the action of H. In addition, we prove a similarity of the Schur-Weyl type duality between the group algebra k [ S e (n) ] of Sergeev group S e (n) and a super pseudoalgebra. [ABSTRACT FROM AUTHOR]

Let A be an associative ring and M a finitely generated projective A-module. We introduce a category RBS (M) and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories RBS (M) naturally arise as generalisations of the exit path ∞ -category of the reductive Borel–Serre compactification of a locally symmetric space, and one of our main techniques is to find purely categorical analogues of some familiar structures in these compactifications. [ABSTRACT FROM AUTHOR]

We prove analogs of A. Selberg's result for finitely generated subgroups of \operatorname {Aut}(A) and of Engel's theorem for subalgebras of \operatorname {Der}(A) for a finitely generated associative commutative algebra A over an associative commutative ring. We prove also an analog of the theorem of W. Burnside and I. Schur about local finiteness of torsion subgroups of \operatorname {Aut}(A). [ABSTRACT FROM AUTHOR]

The Drazin inverse is connected with the notion of index and core-nilpotent decomposition whenever it is discussed in the context of ring of matrices over complex field. In the absence of Drazin inverse for a given element from an arbitrary associative ring (not necessarily with unity), in this paper, the notion of right (left) core-nilpotent decomposition has been introduced and established its relations with right (left) π -regular property. In fact, the class of such decomposition has been characterized. In case of regular ring, observed that an element is right (left) π -regular if and only if it has a right (left) core-nilpotent decomposition. In the process, several properties of sharp order in an associative ring are studied and with the help of the same, new characterizations of Drazin inverse over an associative ring are obtained and the relation between core-nilpotent decomposition and the Drazin inverse is obtained. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity. A unital left R-module M is said to be distributive if for every submodules S, T and U of M, the equality S ∩ (T + U) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for a direct sum of left R-modules to be distributive. This condition is given by the notion of splitting of submodules of the direct sum and the proof uses the notion of orthogonality, where both notions are discussed and revisited. [ABSTRACT FROM AUTHOR]

In this paper, we define and study (co)homology theories of a compatible associative algebra. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures on a vector space. Then we define the cohomology of a compatible associative algebra A and as applications, we study extensions, deformations and extensibility of finite order deformations of A. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras. [ABSTRACT FROM AUTHOR]

In the study of pre-Lie algebras, the concept of pre-morphism arises naturally as a generalization of the standard notion of morphism. Pre-morphisms can be defined for arbitrary (not-necessarily associative) algebras over any commutative ring k with identity, and can be dualized in various ways to generalized morphisms (related to pre-Jordan algebras) and anti-pre-morphisms (related to anti-pre-Lie algebras). We consider idempotent pre-endomorphisms (generalized endomorphisms, anti-pre-endomorphisms). Idempotent pre-endomorphisms are related to semidirect-product decompositions of the sub-adjacent anticommutative algebra. [ABSTRACT FROM AUTHOR]

A pseudo-Euclidean non-associative algebra (g , •) is a finite dimensional algebra over a field K that has a metric, i.e., a bilinear, symmetric, and non-degenerate form 〈 , 〉. The metric is considered L-invariant (resp. R-invariant) if all left multiplications (resp. right multiplications) are skew-symmetric. The metric is called associative if 〈 u • v , w 〉 = 〈 u , v • w 〉 for all u , v , w ∈ g. These three notions coincide when g is a Lie algebra and in this case g endowed with the metric is known as a quadratic Lie algebra. This paper provides a complete description of L-invariant, R-invariant, or associative pseudo-Euclidean metrics on left Leibniz algebras over a commutative field of characteristic zero. It shows that a left Leibniz algebra with an associative metric is also right Leibniz and can be obtained easily from its underlying Lie algebra, which is a quadratic Lie algebra. Additionally, it shows that at the core of a left Leibniz algebra endowed with a L-invariant or R-invariant metric, there are two Lie algebras with one quadratic and the left Leibniz algebra can be built from these Lie algebras. We derive many important results from this complete description. Finally, the paper provides a list of left Leibniz algebras with an associative metric up to dimension 6, as well as a list of left Leibniz algebras with an L-invariant metric, up to dimension 4, and R-invariant metric up to dimension 5. [ABSTRACT FROM AUTHOR]

Let D be an alternative division ring with characteristic different from two. The purpose of this paper is to characterize additive mappings f , g : D → D satisfying certain identities studied by Vukman, Brešar, and Catalano previously on an associative division ring. We also present some new identities concerning Lie and Jordan products, and provide a complete description of commuting maps on octonion algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the one-sided generalized (α, β)-reverse derivation of a ring R. Let R be a semiprime ring, ϱ be a non-zero ideal of R, α be an epimorphism of ϱ, β be a homomorphism of ϱ (α be a homomorphism of ϱ, β be an epimorphism of ϱ) and γ: ϱ → R be a non-zero (α, β)-reverse derivation. We show that there exists F: ϱ → R, an l-generalized (α, β)-reverse derivation (an r-generalized (α, β)-reverse derivation) associated with γ iff F(ϱ), γ(ϱ) ⊂ CR(ϱ) and F is an r-generalized (β, α)-derivation (an l-generalized (β, α)-derivation) associated with (β, α)-derivation γ on ϱ. This theorem generalized the results of A. Aboubakr and S. Gonzalez proved in [1, Theorem 3.1 and Theorem 3.2 ]. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and Id(R) and K(R) denote the set of idempotents and full elements of R respectively. The notion of weakly f-clean rings where element r can be written as r = f +e or r = f-e, e ∈ Id(R) and f ∈ K(R) was introduced. Different properties of weakly f-clean rings were studied. It was shown that a left quasi-duo ring R is weakly clean if and only if R is a weakly f-clean ring. Finally, it was shown that the ring of skew Hurwitz series T = (HR; α) where α is an automorphism of R is weakly f-clean if and only if R is weakly f-clean. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and M a unitary left R-module. An R-module M is said to be uniserial if its submodules are linearly ordered by inclusion. A serial module is a direct sum of uniserial modules. In this paper, we bring our modest contribution to the open problem listed in the book of Alberto Facchini "Module Theory" which states that: is any direct summand of a serial module serial? The answer is yes for particular rings and R-modules. [ABSTRACT FROM AUTHOR]

Several results about one-sided duo rings and duo rings are generalized from the case of unital rings to the case of arbitrary associative rings in this paper. For example, the characterization of one-sided duo rings and duo rings is given, and it is shown that all idempotents of a duo ring commute with all elements of the ring and that every prime ideal is completely prime. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. Let M be a right R-module. A right R-module N is called pseudo nonessential M-principally projective (briefly, pseudo NM-principally projective) if, for each s ∈ S with s(M) ⊄e M, any R-epimorphism from N to s(M) can be lifted to an R-homomorphism from N to M. M is called pseudo nonessential quasi-principally projective (briefly, pseudo NQ-principally projective) if, it is pseudo NM-principally projective. In this paper, we give some characterizations and properties of pseudo NQ-principally projective modules. [ABSTRACT FROM AUTHOR]

There has been considerable interest in recent decades in questions of random generation of finite and profinite groups and finite simple groups in particular. In this paper, we study similar notions for finite and profinite associative algebras. Let k=Fq$k=\mathbb {F}_q$ be a finite field. Let A$A$ be a finite‐dimensional, associative, unital algebra over k$k$. Let P(A)$P(A)$ be the probability that two elements of A$A$ chosen (uniformly and independently) at random will generate A$A$ as a unital k$k$‐algebra. It is known that if A$A$ is simple, then P(A)→1$P(A) \rightarrow 1$ as |A|→∞$|A| \rightarrow \infty$. We extend this result to a large class of finite associative algebras. For A$A$ simple, we find the optimal lower bound for P(A)$P(A)$ and we estimate the growth rate of P(A)$P(A)$ in terms of the minimal index m(A)$m(A)$ of any proper subalgebra of A$A$. We also study the random generation of simple algebras A$A$ by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let A$A$ be a profinite algebra over k$k$. We show that A$A$ is positively finitely generated if and only if A$A$ has polynomial maximal subalgebra growth. Related quantitative results are also established. [ABSTRACT FROM AUTHOR]

In this article we introduce and study the concepts of uncountably generated Krull dimension and uncountably generated Noetherian dimension of an R-module, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension and Noetherian dimension. They respectively rely on the behavior of descending and ascending chains of uncountably generated submodules. It is proved that a quotient finite dimensional module M has uncountably generated Krull dimension if and only if it has Krull dimension, but the values of these dimensions might differ. Similarly, a quotient finite dimensional module M has uncountably generated Noetherian dimension if and only if it has Noetherian dimension. We also show that the Noetherian dimension of a quotient finite dimensional module M with uncountably generated Noetherian dimension β is less than or equal to ω¹ + β, where ω1 is the first uncountable ordinal number. [ABSTRACT FROM AUTHOR]

In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative). [ABSTRACT FROM AUTHOR]

Suppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra $L_R(E)$ is simple if and only if R is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra. [ABSTRACT FROM AUTHOR]

The reversible property of rings was introduced by Cohn and has important generalizations in noncommutative ring theory. In this paper, reversibility of rings is investigated in relation with quasinilpotents and idempotents, and our argument is spread out based on this. We call a ring RQnil e-reversible if for any a , b ∈ R , being a ⁢ b = 0 implies b ⁢ a ⁢ e ∈ R qnil for a prescribed idempotent e ∈ R , where R qnil denotes the set of all quasinilpotent elements of R. In the first, we determine the set R qnil for some classes of rings to investigate the structure of Qnil e-reversible rings. In the second, we use R qnil to define Qnil e-reversibility of rings. The notion of Qnil e-reversible ring is a proper generalization of that of e-semicommutative ring, Qnil-semicommutative ring, e-reversible ring and right (left) quasi-duo ring. We obtain some relations between a ring and its quotient rings in terms of Qnil e-reversibility. Applications via some ring extensions and examples illustrating our results are provided. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity and M an R-bimodule. We introduce the generalization of r-clean rings called left r-clean R-bimodules, defined without their endomor-phism rings. An R-bimodule M is said to be left r-clean if each element is the sum of a left idempotent and a left regular element of M. We present some properties of the left r-clean R-bimodule. At the end of this paper, we give the suficient and necessary condition for an R-bimodule to form a left r-clean R-bimodule. [ABSTRACT FROM AUTHOR]

We introduce the notion of lifting Baer modules, as a generalization of both Baer and lifting modules and give some of their properties. A module M is called lifting Baer if right annihilator of a left ideal of End(M) lies above a direct summand of M. Also, we define the concepts of r-supplemented and amply r-supplemented modules. It is shown that an amply r-supplemened module M that every supplement submodule, is a direct summand of M, is lifting Baer. The relationships between Baer modules and lifting Baer modules are investigated. Morever, we prove that the endomorphism ring of any lifting Baer module is lifting Baer ring. [ABSTRACT FROM AUTHOR]

Assuming that R is an associative ring with prime ideal P, this paper investigates the commutativity of the quotient ring R=P, as well as the possible forms of generalized derivations satisfying certain algebraic identities on R: We give results on strong commutativity, preserving generalized derivations of prime rings, using our theorems. Finally, an example is given to show that the restrictions on the ideal P are not super uous. [ABSTRACT FROM AUTHOR]

The set is a sub-f-ring of RL, that is, the ring of all continuous real-valued functions on a completely regular frame L. The main purpose of this paper is to continue our investigation begun in [3] of extending ring-theoretic properties in RL to the context of completely regular frames by replacing the ring RL with the ring Cc(L) to the context of zero-dimensional frames. We show that a frame L is a CP-frame if and only if Cc(L) is a regular ring if and only if every ideal of Cc(L) is pure if and only if Cc(L) is an Artin-Rees ring if and only if every ideal of Cc(L) with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring Cc(L)=hi is an Artin-Rees ring for any 2 Cc(L) if and only if every minimal prime ideal of Cc(L) is an Artin-Rees ideal. [ABSTRACT FROM AUTHOR]