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Following a suggestion in a footnote of the paper [1] by Fano, we give a direct proof of the fact that one of the two Fano threefolds of degree 72 in P 38 is isomorphic to the anticanonically embedded weighted projective space P (1 , 1 , 4 , 6) . [ABSTRACT FROM AUTHOR]

In this paper we investigate commuting involution graphs in classical affine Weyl groups. Let W be a classical Weyl group of rank n, with W ˜ its corresponding affine Weyl group. Our main result is that if X is a conjugacy class of involutions in W ˜ , then the commuting involution graph C (W ˜ , X) is either disconnected or has diameter at most n + 2. This bound is known to hold for types A ˜ n and C ˜ n , so the main work of this paper is to prove the theorem for types B ˜ n and D ˜ n. [ABSTRACT FROM AUTHOR]

This article is intended to indicate and discuss some errors in N. Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”. [ABSTRACT FROM AUTHOR]

We analyze the proof of the main result of a paper by Cuadra, Etingof and Walton, which says that any action of a semisimple Hopf algebraHon thenth Weyl algebraA = An(K) over a fieldKof characteristic 0 factors through a group algebra. We verify that their methods can be used to show that any action of a semisimple Hopf algebraHon an iterated Ore extension of derivation typein characteristic zero factors through a group algebra. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, S ⊆ R be a multiplicative set, and M be an R-module. We say that a submodule N of M with (N : R M) ∩ S = ∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules. [ABSTRACT FROM AUTHOR]

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms. [ABSTRACT FROM AUTHOR]

In the present paper corrected versions of the statements in the paper "Description of finite-dimensional inner Rickart and Baer Jordan algebras" by F.N. Arzikulov and U.I. Khakimov are given. In particular, it is shown that for any Jordan algebra J with an idempotent p and an associative degenerate radical D such that J = F p + ̇ D , J is an inner RJ-algebra if and only if, for any nonzero a ∈ D , a 2 = 0 and p(pa) = pa. Also, other equivalent conditions when a Jordan algebra J is an inner RJ-algebra are given. As for finite-dimensional nilpotent Jordan algebras, there is not a nilpotent inner RJ-algebra (and hence inner BJ-algebra) except the finite-dimensional Jordan algebra the square of each element of which is zero. [ABSTRACT FROM AUTHOR]

In a previous paper we have compared the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. In the present paper, we compare the Hochschild cohomology groups of finite dimensional monomial algebras under gluing two arrows. [ABSTRACT FROM AUTHOR]

This paper is concerned with the construction of a small, but non-trivial, example of a polynomial identity algebra, which we call the Jackson algebra, that will be used in sequels to this paper to study non-commutative arithmetic geometry. In this paper this algebra is studied from a ring-theoretic and geometric viewpoint. Among other things it turns out that this algebra is a "non-commutative family" of central simple algebras and thus parameterizes Brauer classes over extensions of the base. [ABSTRACT FROM AUTHOR]

Let be a proper closed subset in the projective n-space over a field k of characteristic zero and let C(Y) be its affine cone. Let be the homogeneous defining ideal of Y with homogeneous maximal ideal. In this paper, we find out a lower bound of de Rham depth of C(Y) in terms of. [ABSTRACT FROM AUTHOR]

AbstractIn a recent paper, Baker and Oliver Lorscheid showed that Descartes’s Rule of Signs and Newton’s Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebraic structures, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions for idylls. Such extensions have appeared for semirings with negation symmetries in the work of Akian-Gaubert-Guterman and for hypergroups and hyperfields in the work of Bowler-Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well. [ABSTRACT FROM AUTHOR]

In a finite group G, ρ (G) denotes the product of element orders of G. Let p , q , r , s and t be prime numbers. Recently, it was proved that C p 3 , Cpq, C p 2 q and Cpqrs are characterizable by the product of element orders, and if ρ (G) = ρ (C pqr) , then G ≅ A 5 or C pqr . In this paper, we continue this work and we show that C p × A 5 (for p > 5) and Cpqrst are characterizable by the product of element orders. In the rest of the paper, we show that if ρ (G) = ρ (C pqr) s ρ (C s) p 2 q , then (p , q , r , s) = (2 , 3 , 5 , 11) and G ≅ L 2 (11) . This shows that the structure of ρ (G) uniquely determined L 2 (11) and so L 2 (11) is the only group with this form of ρ (G) . [ABSTRACT FROM AUTHOR]

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : (N μ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q1 and Q2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form (α , α , γ) . [ABSTRACT FROM AUTHOR]

A theorem in the article “Characterization of Monoids by Properties of Generators,” written by Kilp and Knauer, states that amalgamated coproducts of generators are generators in the categoryAct-S of right S-acts over a monoid S. In this note we give a counterexample to deny this theorem. However, we show that the results in that article based on the mentioned theorem still remain valid. Moreover, an incorrect proposition in the monographMonoids, Acts and Categoriesby Kilp et al. which states that coamalgamated product of cogenerators are cogenerators is denied by a counterexample. [ABSTRACT FROM PUBLISHER]

Let o (G) be the average of the element orders of a finite group G. A research topic concerning this quantity is understanding the relation between o (G) and o (H) , where H is a subgroup of G. Let N be the class of finite nilpotent groups and let L (G) be the subgroup lattice of G. In this paper, we show that the set { o (G) o (H) | G ∈ N , H ∈ L (G) } is dense in [ 0 , ∞) . Other density results are outlined throughout the paper. [ABSTRACT FROM AUTHOR]

In this paper, we show that the cover ideals of theta graphs are normal square-free monomial ideals, and also satisfy the strong persistence property. Moreover, we re-prove a well-known result which is related to the cover ideals of bipartite graphs. We finally terminate this paper by stating some theorems which introduce some classes of normal monomial ideals. [ABSTRACT FROM AUTHOR]

In [2003], without giving a detailed proof, Yamauchi provided a formula to calculate the genus of a certain family of smooth complete intersection algebraic curves. That formula is used extensively by Beshaj, Shaska, and Shor to study the algebraic curves for which their Jacobian has superelliptic components. In this note, we determine the correct version of the genus formula with an algebraic proof. Then, we show that the Yamauchi's formula works only under certain conditions. [ABSTRACT FROM AUTHOR]

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. [ABSTRACT FROM AUTHOR]

In this paper, we progress in the general study of rational functional identities in the setting of division rings. The main theorem in the paper states that the only additive maps f and g satisfying that f (x) = − x n g (x − 1) , for n = 3 or 4, are the zero maps. In addition, we provide an alternative proof to the result proved by Dar and Jing regarding the classification of additive maps f and g satisfying that f (x) = − x 2 g (x − 1) . [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 present paper we study the Jordan counterparts of Rickart and Baer * -algebras, i.e., inner RJ-algebras and inner BJ-algebras and prove that a nilpotent Jordan algebra which has no square root nilpotent elements is an inner RJ-algebra. Also we explain that a nilpotent Jordan algebra that has no nilpotent elements with a square root b such that b 3 ≠ 0 is not an inner RJ-algebra if there exists a nonzero element a such that a 2 ≠ 0 . As a main result of the paper we give a description of a finite-dimensional inner RJ-algebra A , isomorphic to R + ̇ N , with a nilradical N and a finite-dimensional inner BJ-algebra with a nilradical N. [ABSTRACT FROM AUTHOR]

This very short correction notes a gap in an argument of an earlier paper, and also provides a theorem of similar flavor to the main result of that paper. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) the edge ring k [ G ] is k [ x 1 , ... , x n ] / I (G) , where n = | V | and I(G) is generated by { x i x j ; { i , j } ∈ E } . The conjecture we treat is the following. Conjecture 1. If k [ G ] has a 2-linear resolution, then the projective dimension of K [ G ] , pd (k [ G ]) , equals the maximal degree of a vertex in G. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq [1]. That there are counterexamples was noted already by Moradi and Kiani [9]. By interpreting k [ G ] as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds. [ABSTRACT FROM AUTHOR]

A ring idempotent e ∈ R is said to be quarter-central (or q-central for short) if e R (1 − e) R e = 0 . If all idempotents in a ring R are q-central, R is said to be a quarter-abelian ring (or more simply, a q-abelian ring). In this paper, we show that such a ring R is characterized by the property that [ e , R ] [ e , R ] = 0 for all idempotents e ∈ R , where [ x , y ] denotes the additive commutator x y − y x . For any nonzero ring S and any integer n ≥ 3 , we show that the ring T n (S) of n × n upper triangular matrices over S is not q-abelian. On the other hand, T 2 (S) is q-abelian iff S is abelian (in the classical sense that all idempotents are central in S ). A final section of this paper relates q-central idempotents to the notion of exchange rings and the study of regular, unit-regular, and strongly regular elements in arbitrary rings. From the viewpoint of currently prevailing generalizations of abelian rings in the literature, q-abelian rings are situated between the class of semiabelian rings and the class of strongly IC rings. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Recently various functions are defined related to the set of order elements of G and using these functions, some interesting criteria for solvability, nilpotency, supersolvability and etc., are obtained. In this paper, we continue this work and for a finite group G, we consider the function m (G) = ∑ g ∈ G 1 / o (g) , where o(g) denotes the order of g ∈ G . In this paper, we prove that if m (G) < m (A 5) , m (G) < m (A 4) , m (G) < m (S 3) , m (G) < m (Q 8) or m (G) < m (C 2 × C 2) , then G is solvable, supersolvable, nilpotent, abelian or cyclic, respectively. [ABSTRACT FROM AUTHOR]

This work is devoted to the complete classification of solvable Leibniz algebras with the naturally graded quasi-filiform non-Lie nilradical. It should be noted that the class of naturally graded quasi-filiform Leibniz algebras is sufficiently large, and solvable Leibniz algebras with the nilradical which is isomorphic to several of these algebras are classified in a series of works. This paper considered all omitted cases and completed the problem of classifying solvable Leibniz algebras with naturally graded quasi-filiform non-Lie nilradical. At the end of the paper we combine all results obtained to date, and provide a complete list of solvable Leibniz algebras, whose nilradical is a naturally graded quasi-filiform Leibniz algebra. [ABSTRACT FROM AUTHOR]

Köthe's classical problem posed by G. Köthe in 1935 asks to describe the rings R such that every left R-module is a direct sum of cyclic modules (these rings are known as left Köthe rings). Köthe, Cohen and Kaplansky solved this problem for all commutative rings (that are Artinian principal ideal rings). During the years 1962 to 1965, Kawada solved Köthe's problem for basic finite-dimensional algebras. But, so far, Köthe's problem was open in the non-commutative setting. Recently, in the paper [Several characterizations of left Köthe rings, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. (2023) (to appear)]], we classified left Köthe rings into three classes one contained in the other: left Köthe rings, strongly left Köthe rings and very strongly left Köthe rings, and then, we solved Köthe's problem by giving several characterizations of these rings in terms of describing the indecomposable modules. In this paper, we will introduce the Morita duals of these notions as left co-Köthe rings, strongly left co-Köthe rings and very strongly left co-Köthe rings, and then, we give several structural characterizations for each of them. [ABSTRACT FROM AUTHOR]

In regard to two recent publications in the Mediterranean J. Math. (2021) and Forum Math. (2021) related to fully and characteristically inert socle-regularity, respectively, we define and study the so-called weakly characteristically inert socle-regular groups. In that aspect, as a culmination of the investigations of this sort, some more global results are obtained and, moreover, some new concrete results concerning the weakly fully inert socle-regular groups, defined as in the firstly mentioned above paper, are also established. In particular, we prove that all torsion-complete groups are characteristically inert socle-regular, which encompasses an achievement from the secondly mentioned paper and completely settles the problem posed there about this class of groups. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, Wolbachia infection in a bisexual and diploid population with a fixed cytoplasmic incompatibility rate w and maternal transmission rate d is studied as an evolution algebra. As the cytoplasmic incompatibility (CI) of the population causes deaths in the offspring, the evolution algebra of this model is not baric, and is a dibaric algebra if and only if the cytoplasmic incompatibility rate w is 1 and d=1 . The idempotent elements are given in terms of d and w. Moreover, this algebra has no absolute nilpotent elements when CI expression w≠1 . [ABSTRACT FROM AUTHOR]

AbstractLet R be a commutative ring with identity and R′ be the integral closure of R. In this paper, we show that if R is an affine algebra over a field K, then every regular ideal of R′ is finitely generated, i.e., R′ is an r-Noetherian ring. We also study when the integral closure of an affine algebra is Noetherian. First we show that if R is a Krull ring such that R/P is a Noetherian domain for each minimal regular prime ideal P of R, then R is an r-Noetherian ring, which is a generalization of Nishimura’s result. As an application of this result, we prove that if R is an r-Noetherian ring with reg-dim R≤2 , then R′ is an r-Noetherian ring. We finally construct a couple of r-Noetherian rings, e.g., an r-Noetherian ring R that is not Noetherian and reg-dim R=∞ or reg-dim R=n≤ dim R=n+m−1 for arbitrary positive integers n, m. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we generalize the classical method used to classify nilpotent low-dimensional Lie algebras to serve for nilpotent Bol algebras. The algebraic classification of the nilpotent Bol algebras up to dimension four is given. [ABSTRACT FROM AUTHOR]

AbstractLet G be a locally compact group. We denote by Sub(G) the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper we establish that the subset SubLE(G) of closed locally elliptic subgroups of G is Chabauty-closed if and only if the set compn(G) consisting of the n-tuples of elements of G contained in a common compact subgroup, is closed in the Cartesian product Gn, for every nonnegative integer n. Moreover, we prove that the subspace SubLE(G) is Chabauty-closed in the case when the group G contains an open normal locally elliptic subgroup. [ABSTRACT FROM AUTHOR]

For an irreducible character χ of a finite group G, the codegree of χ is defined as | G : ker (χ) | / χ (1) . In this paper, we characterize finite p-groups with three character codegrees. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete classification of torsion pairs in m-cluster categories of type D when m is odd, denoted by C D n m , via a bijection to combinatorial objects called Ptolemy diagrams of type D. As applications, we classify m-rigid subcategories of C D n m , which gives Jacquet-Malo's classification of m-cluster tilting subcategories of C D n m . When m = 1, we generalizes the work of Holm, Jørgensen and Rubey [14] for the classification of torsion pairs in cluster categories of type Dn. Communicated by Steffen Oppermann [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explain a method on the generalization of the Bertini-type theorem on standard graded rings to the non-standard graded case of certain type. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, MS, is a flat RS-module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s ∈ S . While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the "S-perfectness" for various ring constructions, which allows the construction of more interesting examples. [ABSTRACT FROM AUTHOR]

Let Γ be a torsion-free cancellative monoid and let R = ⊕ α ∈ Γ R α be a graded integral domain. In this paper, we define the concept of homogeneous star-operations, which is a generalization of that of star-operations, and apply some properties of star-operations. Also, we define the concept of homogeneous divisor classes, which is the graded integral domain version of that of divisor classes, and study some equivalent conditions for the set of homogeneous divisor classes H D (R) of R to be a group. [ABSTRACT FROM AUTHOR]

Let M (S ; Λ ; P) denote a Rees I × Λ matrix semigroup over a semigroup S in which I is a singleton. In this paper we describe the right regular representation of M (S ; Λ ; P) , and find connection between properties of S and M (S ; Λ ; P) . We also prove embedding theorems on Rees matrix semigroups M (S ; Λ ; P) . [ABSTRACT FROM AUTHOR]

Let (X , ≤) be a locally finite pre-ordered set and R be a commutative ring with unity. In this paper we apply the theory of zero product determined algebras to show that each linear map on the incidence algebra I (X , R) which is derivable at zero is a generalized derivation and every local derivation on I (X , R) is a derivation. [ABSTRACT FROM AUTHOR]

By means of the maximal left ring of quotients, this paper proves that every n-Lie m-derivation on a triangular ring can be resolved into a sum of an extremal n-derivation and an n-additive central mapping. As applications, n-Lie m-derivations on some classical triangular rings are characterized. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a given commutative Noetherian ring R which satisfies the condition of the first problem of R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145–164]. In this paper, we prove that a also satisfies the condition of his second problem in the same article. We also provide an example to show that the converse statement does not hold in general. [ABSTRACT FROM AUTHOR]

Let p be a prime and F p be a finite field of p elements. Let F p G denote the group algebra of the finite p-group G over the field F p and V (F p G) denote the group of normalized units in F p G . Suppose that G is a finite p-group given by a central extension of the form 1 → Z p n × Z p m → G → Z p × ⋯ × Z p → 1 and G ′ ≅ Z p , n , m ≥ 1 and p is odd. In this paper, the structure of G is determined. And the relations of V (F p G) p l and G p l , Ω l (V (F p G)) and Ω l (G) are given. Furthermore, there is a direct proof for V (F p G) p ∩ G = G p . [ABSTRACT FROM AUTHOR]

The notion of double extension for symplectic Lie algebras was introduced by Medina and Revoy in 1985. In 2021, Valencia gave a condition for symplectic Lie algebras which are obtained through double extension to admit compatible left-symmetric structures. In this paper, we formulate the notion of double extension for left-symmetric structures and give a condition for left-symmetric structures to be obtained through double extension. Moreover, we prove that right nilpotent left-symmetric structures on some solvable Lie algebras are obtained through double extension. We also give a condition for left-symmetric structures which are obtained through double extension to be complete. [ABSTRACT FROM AUTHOR]

Since the existing tropical cryptographic protocols are either susceptible to the Kotov-Ushakov attack and its generalization, or to attacks based on tropical matrix periodicity and predictive behavior, several attempts have been made to propose protocols that resist such attacks. Despite these attempts, many of the proposed protocols remain vulnerable to attacks targeting the underlying hidden problems, one of which we call the tropical two-sided discrete logarithm with shift. An illustrative case is the tropical Stickel protocol, which, when formulated with a single monomial instead of a polynomial, becomes susceptible to attacks based on solutions of the above mentioned tropical version of discrete logarithm. In this paper we will formally introduce the tropical two-sided discrete logarithm with shift, discuss how it is solved, and subsequently demonstrate an attack on a key exchange protocol based on the tropical semiring of pairs. This particular protocol is compromised due to the existence of efficient (albeit heuristic) solution of the tropical two-sided logarithm problem, and this highlights the ongoing challenges in search of a "good" key exchange protocol in tropical cryptography. [ABSTRACT FROM AUTHOR]

In this paper, Whittaker modules are studied for a subalgebra q ϵ of the N =2 superconformal algebra. The Whittaker modules are classified by central characters. Additionally, criteria for the irreducibility of the Whittaker modules are given. [ABSTRACT FROM AUTHOR]

In this paper we study the modular structure of the Foulkes module H (2 m) of the symmetric group S 2 m acting on set partitions of a set of size 2m into m sets each of size 2, defined over a field of positive characteristic p. We aim to understand the decomposition of the tensor powers of H (2 m) . In particular, we study the asymptotics of the non-projective part of tensor powers of these modules by computing the gamma invariant as defined by Dave Benson and Peter Symonds. [ABSTRACT FROM AUTHOR]

Let A be an associative superalgebra endowed with a pseudoautomorphism p. In this paper we generalize the Wedderburn-Malcev Theorem in this setting and we prove that the sequence of p-codimensions of A is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and the algebra of 2 × 2 upper-triangular matrices with suitable pseudoautomorphisms. [ABSTRACT FROM AUTHOR]

AbstractOur aim in this paper is to initiate the study of exponent semigroups for rational matrices. We prove that every numerical semigroup is the exponent semigroup of some rational matrix. We also obtain lower bounds on the size of such matrices and discuss the related class of power-integral matrices. [ABSTRACT FROM AUTHOR]

AbstractAn element a in a ring R is called a quasi-idempotent if a2=ka for some central unit k of R. A ring R is called quasi-exchange if for any element x of R, there exists a∈R with a2=ka∈Rx and k−a∈R(k−x) , where k is a central unit of R. Quasi-exchange rings are a generalization of exchange rings and quasi-clean rings. This paper studies the properties of quasi-exchange rings and the relationship among quasi-exchange rings, exchange rings and quasi-clean rings. An equation xa−fx=1 is used to characterize the quasi-exchange ring and we prove that quasi-exchange rings are left-right symmetric. [ABSTRACT FROM AUTHOR]