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In this paper, we give a necessary and sufficient condition for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroup using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also discuss nice extensions for simplicial affine semigroups, a concept which is motivated by nice extensions of numerical semigroups. [ABSTRACT FROM AUTHOR]

The famous Kurosh Problem for division rings asks whether a finitely generated algebraic division ring is centrally finite. This longstanding problem posed in 1941 by Kurosh remains without answer up to now. In this paper, we study some questions related to this problem for some classes of division rings. [ABSTRACT FROM AUTHOR]

A semigroup is called a Hua semigroup if any mapping h : S → S satisfying (∀ a , b ∈ S) h (a b) = h (a) h (b) or h (b) h (a) is either a homomorphism or an anti-homomorphism. We use this name for such a semigroup since it is relevant to a well-known result in ring theory, which is established by L. K. Hua in 1949. C. M. Reis and H. J. Shyr first translated Hua's result from rings to semigroups in 1977. In this paper, elementary characterizations, a hierarchy and closure properties of the class of Hua semigroups and its four subclasses are given. We show that every cancellative semigroup is a Hua semigroup, which generalizes Reis and Shyr's result. [ABSTRACT FROM AUTHOR]

In this paper, we give a necessary and sufficient condition for a graded algebra to be a McCoy ring. We then classify all McCoy rings of finite dimensional path algebras and truncated algebras over field K. This leads a new way of constructing McCoy rings. [ABSTRACT FROM AUTHOR]

Recall that a commutative ring R is said to be a normal ring if it is reduced and every two distinct minimal prime ideals are comaximal. A finitely generated reduced R-module M is said to be a normal module if every two distinct minimal prime submodules are comaximal. The concepts of normal modules and locally torsion free modules are different, whereas they are equal in theory of commutative rings. We give many properties and examples of normal modules, we use them to characterize locally torsion free modules and Baer modules. Also, we give the topological characterizations of normal modules. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. Firstly, to give short proofs of the celebrated results of Rudd (1970), De Marco (1972), Brookshear (1977), Neville (1990), and Vechtomov (1996) with the help of some well-known results in ring theory. Secondly, to establish new algebraic characterizations for when C(X) is von Neumann regular, semihereditary, or a Bézout ring. The notions of the greatest common divisor and least common multiple in the ring C(X) are studied. Finally, the space X, for which every idempotent of the classical quotient ring of C(X) is actually an element of C(X) itself, is completely determined. [ABSTRACT FROM AUTHOR]

The Nagata ring R(X) and the Serre’s conjecture ring R⟨X⟩ are two localizations of the polynomial ring R[X] at the polynomials of unit content and at the monic polynomials, respectively. In this paper, we contribute to the study of Prüfer conditions in R(X) and R⟨X⟩. In particular, we solve the four open questions posed by Glaz and Schwarz in Section 8 of their survey paper [38] related to the transfer of Prüfer conditions to these two constructions. [ABSTRACT FROM AUTHOR]

We study the ring of polyfunctions over Z / n Z. The ring of polyfunctions over a commutative ring R with unit element is the ring of functions f : R → R which admit a polynomial representative p ∈ R [ x ] in the sense that f (x) = p (x) for all x ∈ R. This allows to define a ring invariant s which associates to a commutative ring R with unit element a value in N ∪ { ∞ }. The function s generalizes the number theoretic Smarandache function. For the ring R = Z / n Z we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number Ψ (n) of polyfunctions over Z / n Z. We also investigate algebraic properties of the ring of polyfunctions over Z / n Z. In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive formulas for the size of the ring of polyfunctions in several variables over Z / n Z , and we compute the number of polyfunctions which are units of the ring. [ABSTRACT FROM AUTHOR]

A quasi-order on a set S is a binary, reflexive and transitive relation on S. In [3], Fakhruddin introduced the notion of (totally) quasi-ordered fields and showed that each such field is either an ordered field or else a valued field. Hence, quasi-ordered fields are very well suited to treat ordered and valued fields simultaneously. The aim of the present paper is to prove that an analogous dichotomy holds for commutative rings with 1 as well. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with unity 1 and let a,b,c∈R. In this paper, several characterizations for hybrid (b,c)-inverses of a are given. Also, the hybrid (b,c)-inverse of a is characterized by the group inverse of ab, under certain hypothesis. In particular, existence criteria for the the inverse along an element are obtained. Finally, we get the double commutant property and the reverse order law of annihilator (b,c)-inverses. [ABSTRACT FROM AUTHOR]

In the first part of this paper, we point out some errors in Communications in Algebra, 45(11) (2017), 4631–4645 and then we give a correct version of the errant results. The second part presents our contribution in the study of two-sided α-derivations satisfying certain identities over near-rings. [ABSTRACT FROM AUTHOR]

Let R be a noncommutative prime ring, I a nonzero left ideal of R, L a non-central Lie ideal of R, U the left Utumi quotient ring of R and C = Z (U) the extended centroid of R. Let G be a generalized derivation of R and g be a derivation of R and m , n , r , k fixed positive integers. In the present paper, we describe the structure of R and all possible forms of G and g in the following cases: [ G (x m) x n + x n g (x m) , x r ] k = 0 , for all x ∈ I ; char (R) ≠ 2 and [ G (x m) x n + x n g (x m) , x r ] k = 0 , for all x ∈ L. [ABSTRACT FROM AUTHOR]

Mutation of τ-tilting modules is a basic operation to construct a new support τ-tilting module from a given one by replacing a direct summand. The aim of this paper is to give a positive answer to the question posed in [2, Question 2.31] about mutation of τ-tilting modules. [ABSTRACT FROM AUTHOR]

With the present note the authors intend to correct a mistake in the paper Semiring and semimodule issues in MV-algebras, Communications in Algebra 41:3 (2013), 1017-1048, by the same authors. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a ring R is left coherent if and only if the class 풜bs of absolutely pure left R-modules is a covering class. [ABSTRACT FROM AUTHOR]

Let (R,픪) be a Cohen-Macaulay local ring of positive dimension d and infinite residue field. Let I be an 픪-primary ideal and J a minimal reduction of I. In this paper, we show that . This answer to a question that made by Rossi and Swanson in [15, Question 4.6]. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the ring of Colombeau full generalized quaternion, and we study its algebraic and topological properties. We prove that the ring of Colombeau full generalized quaternion is Gelfand, normal, duo and some related properties. Moreover, we study the essential ideals of this ring. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity. [ABSTRACT FROM AUTHOR]

In this paper, we study the graded Thierrin radical and the classical Thierrin radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two versions of this graded radical, the graded Thierrin and the large graded Thierrin radical. We establish several characterizations of the graded Thierrin radical and prove that the largest homogeneous ideal contained in the classical Thierrin radical of a graded ring coincides with the large graded Thierrin radical of that ring. [ABSTRACT FROM AUTHOR]

Let R be a semiprime ring and I a nonzero ideal of R. A map F:R→R is called a multiplicative generalized derivation if there exists a map d:R→R such that F(xy) = F(x)y+xd(y), for all x,y∈R. In the present paper, we shall prove that R contains a nonzero central ideal if any one of the following holds: i) ii) iii) F is SCP on I, iv) F(u)∘F(v) = u∘v, for all u,v∈I. [ABSTRACT FROM AUTHOR]

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree, so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, z-Poisson algebras, Gerstenhaber algebras, and Schouten algebras among other classes of algebras. The present paper is devoted to the study of structure of Poisson color algebras of degree g0, where g0 is some element of the grading group G such that g0 = 0 or 4g0≠0, and with restrictions neither on the dimension nor the base field, by stating a second Wedderburn-type theorem for this class of algebras. [ABSTRACT FROM AUTHOR]

We study the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. In 1983, Anick proved the fundamental result on approximation of polynomial automorphisms. We obtain similar approximation theorems for symplectomorphisms and Weyl algebra authomorphisms. We then formulate the lifting problem. More precisely, we prove the possibility of lifting of a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory. This paper is a continuation of the study [19]. [ABSTRACT FROM AUTHOR]

Let (R,픪) be a Noetherian local ring. In this paper, we give a new characterization for the universal catenaricity of R and the Cohen-Macaulayness of all its formal fibers, and we study the cofiniteness and annihilators of local cohomology modules. [ABSTRACT FROM AUTHOR]

The rate of a graded moduleMover a standard graded algebraR,rateR(M), is a measure of the growth of the shifts in the minimal graded free resolution ofM. In this paper, we study the rate of Veronese modules ofM. More precisely, it is shown thatfor allc≥1. This extends a result of Aramova et al. [2] As a consequence of this, ifMis generated in degree zero, then, for allc≥max{rateR(M),Rate(R)}. Also, it is shown that, for all integerss,c≥1. In particular case, we give a simple proof to a result of Backelin. [ABSTRACT FROM PUBLISHER]

In this paper we provide conditions under which automorphism-coinvariant modules over a right perfect ring are quasi-projective. [ABSTRACT FROM PUBLISHER]

In this paper, we study third power associative, antiflexible rings satisfying the identity (a,b,ac) = a(a,b,c). We prove that third power associative, antiflexible rings satisfying the identity (a,b,ac) = a(a,b,c) with characteristic ≠2,3 are associative of degree 5. As a consequence of this result, we prove that a third power associative semiprime antiflexible ring satisfying the identity (a,b,ac) = a(a,b,c) is associative. [ABSTRACT FROM AUTHOR]

In 1971, Koehler [11] proved a structure theorem for quasi-projective modules over right perfect rings by using results of Wu-Jans [22]. Later Mohamed-Singh [17] studied discrete modules over right perfect rings and gave decomposition theorems for these modules. Moreover, Oshiro [18] deeply studied (quasi-)discrete modules over general rings. In this paper, we consider that decomposition theorems for H-supplemented modules with the condition (D2) or (D3) over right perfect rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce von Neumann regular modules and give many characterizations of von Neumann regular modules. Further, we investigate the relations between von Neumann regular modules and other classical modules. Finally, we characterize Noetherian von Neumann regular modules. [ABSTRACT FROM AUTHOR]

The notion of a U-module was introduced and thoroughly investigated in [11] as a strict and simultaneous generalization of quasi-continuous, square-free and automorphism-invariant modules. In this paper a right R-module M is called a U*-module if every submodule of M is a U-module, and a ring R is called a right U*-ring if RR is a U*-module. We show that M is a U*-module iff whenever A and B are submodules of M with A≅B and A∩B = 0, A⊕B is a semisimple summand of M; equivalently M = X⊕Y, where X is semisimple, Y is square-free, and X & Y are orthogonal. In particular, a ring R is a right U*-ring iff R is a direct product of a square-full semisimple artinian ring and a right square-free ring. Moreover, right U*-rings are shown to be directly-finite, and if the ring is also an exchange ring then it satisfies the substitution property, has stable-range 1, and hence is stably-finite. These results are non-trivial extensions of similar ones on rings all of whose right ideals are either quasi-continuous or auto-invariant. [ABSTRACT FROM AUTHOR]

Let V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one. [ABSTRACT FROM AUTHOR]

The ring of integer-valued polynomials on an arbitrary integral domain is well studied. In this paper, we initiate and provide motivation for the study of integer-valued polynomials on commutative rings and modules. Several examples are computed, including the integer-valued polynomials over the ring for any commutative ring R and any elements of R, as well as the integer-valued polynomials over the Nagata idealization R(+)M of M over R, where M is an R-module. These examples are motivated by and provide motivation for the study of integer-valued derivatives. [ABSTRACT FROM AUTHOR]

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ℳ of the form ℳ = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic p>0 and let KG be the group algebra of an arbitrary group G over K. It is known that if KG is Lie nilpotent, then its lower as well as upper Lie nilpotency index is at least p+1. The group algebras KG for which these indices are p+1 or 2p or 3p−1 or 4p−2 have already been determined. In this paper, we classify the group algebras KG for which the upper Lie nilpotency index is 5p−3, 6p−4 or 7p−5. [ABSTRACT FROM AUTHOR]

LetRbe an arbitrary ring with identity andM, a rightR-module withZ2(M) the second singular submodule ofM. We call an endomrphismφofM, a t-automorphism ifdefined byis an isomorphism and we callM, a t-automorphism-invariant module if it is invariant under t-automorphisms of its injective hulls. This paper is devoted to investigation of various properties and characterizations of t-automorphism-invariant modules. We show that a moduleMis t-automorphism invariant if and only if every t-isomorphism between two t-essential submodules ofMextends to an endomorphism ofM. [ABSTRACT FROM PUBLISHER]

LetRbe a ring with involution. In this paper, we introduce a new type of generalized inverse called pseudo core inverse inR. The notion of core inverse was introduced by Baksalary and Trenkler for matrices of index 1 in 2010 and then it was generalized to an arbitrary ∗-ring case by Rakić, Dinčić and Djordjević in 2014. Our definition of pseudo core inverse extends the notion of core inverse to elements of an arbitrary index inR. Meanwhile, it generalizes the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for matrices in 2014, to the case of ∗-ring. Some equivalent characterizations for elements inRto be pseudo core invertible are given and expressions are presented especially in terms of Drazin inverse and {1,3}-inverse. Then, we investigate the relationship between pseudo core inverse and other generalized inverses. Further, we establish several properties of the pseudo core inverse. Finally, the computations for pseudo core inverses of matrices are exhibited. [ABSTRACT FROM AUTHOR]

In [1], Bannai presents a fusion condition and uses this to consider central Schur rings (S-rings) over the simple groupsPSL(2,q) whereqis a prime power. In this paper, we concretely describe all such S-rings in terms of symmetric S-rings over cyclic groups. The final section discusses counting these. [ABSTRACT FROM PUBLISHER]

An abstract class of R-Mod is termed Additive Class if it is closed under taking submodules, homomorphic images and finite direct sums. In this paper we study the lattice properties of R-ad, the big lattice of all additive classes in R-mod for R any ring. Also, we study the lattice properties of R-ad when R is a Noetherian or semisimple ring. We describe a characterization of some rings through the additive classes and their properties. We study some properties of additive classes related with the associated linear filters. [ABSTRACT FROM AUTHOR]

Pre-Koszul and Koszul algebras were defined by Priddy [15]. There exist some relations between these algebras and the skew PBW extensions defined in [8]. In [24] we gave conditions to guarantee that skew PBW extensions over fields it turns out homogeneous pre-Koszul or Koszul algebra. In this paper we complement these results defining graded skew PBW extensions and showing that ifRis a finite presented Koszul𝕂-algebra then every graded skew PBW extension ofRis Koszul. [ABSTRACT FROM AUTHOR]

Letℛbe a commutative ring with identity and let𝔄 = Tri(𝒜,ℳ,ℬ) be a triangular algebra consisting of unital algebras𝒜,ℬoverℛand an (𝒜,ℬ)-bimoduleℳwhich is faithful as a left𝒜-module as well as a rightℬ-module. In this paper, we prove that under certain assumptions every nonlinear generalized Lie triple derivationGL:𝔄→𝔄is of the formGL = δ+τ, whereδ:𝔄→𝔄is an additive generalized derivation on𝔄andτis a mapping from𝔄into its center which annihilates all Lie triple products [[x,y],z]. [ABSTRACT FROM AUTHOR]

In this paper, we study properties of isomorphisms of global rings that preserve the standard bases. [ABSTRACT FROM AUTHOR]

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2] and were extensively investigated by Ahn and Anderson [1], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study torsion-theoretic generalizations of singular and nonsingular modules by using the concept ofτ-essential submodule for a hereditary torsion theoryτ. We introduce two new module classes calledτ-singular and non-τ-singular modules. We investigate some properties of these module classes and present some examples to show that these new module classes are different from singular and nonsingular modules. We give a characterization ofτ-semisimple rings via non-τ-singular modules. We prove that ifM∕τ(M) is non-τ-singular for a moduleM, then every submodule ofMhas a uniqueτ-closure. We give some properties of the torsion theory generated by the class of allτ-singular modules. We obtain a decomposition theorem for a stronglyτ-extending module by using non-τ-singular modules. [ABSTRACT FROM AUTHOR]

A principal right ideal of a ring is called uniquely generated if any two elements of the ring that generate the same principal right ideal must be right associated (i.e., if for alla,bin a ringR,aR = bRimpliesa = bufor some unituofR). In the present paper, we study “uniquely generated modules” as a module theoretic version of “uniquely generated ideals,” and we obtain a characterization of a unit-regular endomorphism ring of a module in terms of certain uniquely generated submodules of the module among some other results:End(M) is unit-regular if and only ifEnd(M) is regular and allM-cyclic submodules of a rightR-moduleMare uniquely generated. We also consider the questions of when an arbitrary element of a ring is associated to an element with a certain property. For example, we consider this question for the ringR[x;σ]∕(xn+1), whereRis a strongly regular ring with an endomorphismσbe an endomorphism ofR. [ABSTRACT FROM PUBLISHER]

Automorphism-invariant modules, due to Lee and Zhou, generalize the notion of quasi-injective modules. A module which is invariant under automorphisms of its injective hull is called an automorphism-invariant module. Here we carry out a study of the module which is invariant under nilpotent endomorphisms of its injective envelope, such as modules are called nilpotent-invariant. Many basic properties are obtained. For instance, it is proved that (1) nilpotent-invariant modules have the (C3) property, (2) ifis nilpotent-invariant, thenM1andM2are relative injective. In this paper, we also show that (3) a simple right nilpotent-invariant ringRis either right self-injective orRRis uniform square-free. [ABSTRACT FROM PUBLISHER]

The zero-divisor graphs of commutative rings have been used to build bridges between ring theory and graph theory. Namely, they have been used to characterize many ring properties in terms of graphic ones. However, many results are established only for reduced rings because a zero-divisor graph defined in the classical manner lacks the information on relationship between powers of zero-divisors. The aim of this article is to remedy this situation by introducing a parametrized family of graphs { Γ ¯ i (R) } i ∈ N * , for a ring R, which reveals more of the relationship between powers of zero-divisors as follows: For each i ∈ N * , Γ ¯ i (R) is the simple graph whose vertex set is the set of non-zero zero-divisors such that two distinct vertices x and y are joined by an edge if there exist two positive integers n ≤ i and m ≤ i such that x n y m = 0 with x n ≠ 0 and y m ≠ 0. Our aim is to study in detail the behavior of the filtration { Γ ¯ i (R) } i ∈ N * as well as the relations between its terms. We give answers to several interesting and natural questions that arise in this context. In particular, we characterize girth and diameter of Γ ¯ i (R) and give various examples. [ABSTRACT FROM AUTHOR]

In this study, we find height bounds in the polynomial ring over the field of algebraic numbers to test the primality of an ideal. We also obtain height bounds in the arithmetic Nullstellensatz. We apply nonstandard analysis and hence our constants will be ineffective. [ABSTRACT FROM AUTHOR]

LetRbe a ring containing a nontrivial idempotent with centerZ(R) andℕbe the set of all non-negative integers. A familyof mapsdn:R→R(need not be additive) is said to be a Lie triple higher derivable iffor alla,b,c∈Rand for alln∈ℕ, where(the identity map on R). In the present paper it is shown that ifdnis a Lie triple higher derivable for eachn∈ℕ, then there exists an elementza,b(depending onaandb) in its centerZ(R) such thatfor alln∈ℕ. [ABSTRACT FROM AUTHOR]

A ∗-ringRis called ∗-regular if every principal one-sided ideal ofRis generated by a projection. The paper is devoted to a study of ∗-regularity of ∗-rings. Basic properties of ∗-regular rings are investigated, and some equivalent characterizations on ∗-regular rings, Abelian ∗-regular rings, and *-unit regular rings are provided. We also show that a matrix ringMn(R) is ∗-unit regular if and only ifRis unit regular andis a unit for allxiinR. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate commutativity of ringRwith involution ∗ which admits a derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Finally, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous. [ABSTRACT FROM PUBLISHER]

IfGis a group and if the upper hypercenter,Z, ofGis such thatG∕Zis finite then a recent theorem shows thatGcontains a finite normal subgroupLsuch thatG∕Lis hypercentral. The purpose of the current paper is to obtain a version of this result for subgroupsGofGL(F,A), whenAis an infinite dimensionalF-vector space. [ABSTRACT FROM AUTHOR]