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AbstractOur present work is motivated by several papers which have introduced and studied the notion of nonnil-Noetherian rings. Let A = ⊕α∈G Aα be a commutative ring with unity graded by an arbitrary commutative monoid G. We define A to be graded-nonnil-Noetherian ring if every nonnil graded ideal of A, that is, a graded ideal which is not contained in Nil(A), is finitely generated. The purpose of this paper is to give the graded version of several characterizations, properties and basic results proved in [3], [7] and [4]. We also show the non-triviality of our generalization by giving examples of graded-nonnil-Noetherian rings which are not nonnil-Noetherian and of graded-nonnil-Noetherian rings which are not graded-Noetherian. [ABSTRACT FROM AUTHOR]

Let $ \mathcal {G}_R $ G R denote a unitary one-matching bi-Cayley graph over a finite commutative ring R. In this paper, we give a necessary and sufficient condition for $ \mathcal {G}_R $ G R (respectively, the complement of $ \mathcal {G}_R $ G R and the line graph of $ \mathcal {G}_R $ G R ) to be Ramanujan. [ABSTRACT FROM AUTHOR]

On the Waring's problem for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as sums of kth powers with $ k\in \{2,3,4,5,6,7,8\} $ k ∈ { 2 , 3 , 4 , 5 , 6 , 7 , 8 } in the literature. In this paper, we provide similar conditions for matrices written as sums of kth powers with $ k\in \{9,10,11,12,13,14,15,16\} $ k ∈ { 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 }. [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 this paper we introduce the notion of functional prime ideals in a commutative ring. For a (left) R-module M and a functional ϕ (i.e., an R-linear map ϕ from M to R), an ideal I of R is said to be a ϕ -prime ideal if whenever a ∈ R and m ∈ M such that a ϕ (m) ∈ I , then a ∈ I or ϕ (m) ∈ I . This notion shows its ability to characterize different classes of ideals in terms of functional primeness with respect to specific R-modules. For instance, if the module M is the ideal I itself, then I is ϕ -prime for every ϕ ∈ Hom R (I , R) if and only if I is a trace ideal, and if the module M is the dual of I, then I is ϕ -prime for every ϕ ∈ Hom R (I − 1 , R) if and only if I is a prime ideal of R, or I is a strongly divisorial ideal. Several results are obtained and examples to illustrate the aims and scopes are provided. [ABSTRACT FROM AUTHOR]

AbstractLocal log-regular rings are a certain class of Cohen-Macaulay local rings that are treated in logarithmic geometry. Our paper aims to provide purely commutative ring theoretic proof of several ring-theoretic properties of local log-regular rings such as an explicit description of a canonical module, and the finite generation of the divisor class group. [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]

We investigate the Möbius groups theory in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero-divisors. In this paper, we first generalize classical conjugacy classification to bicomplex analysis. Then we have a discussion of the iterates of a bicomplex Möbius transformation and study the attractive and repulsive fixed points in bicomplex setting. Finally, we shall prove two useful results concerning fixed point. [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 (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]

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]

AbstractLet k be an arbitrary field and let b>1,n>1 and a be three positive integers. In this paper we explicitly describe a minimal S-graded free resolution of the semigroup algebra k[S] when S is a generalized repunit numerical semigroup, that is, when S is the submonoid of N generated by {a1,a2,…,an} where a1=∑j=0n−1bj and ai−ai−1=a bi−2, i=2,…,n, with gcd(a,a1)=1. [ABSTRACT FROM AUTHOR]

Let S be an algebra over a commutative ring R, and let σ be an automorphism on S. In this paper, we investigate the notion of generalized σ-derivations on modules, which is an extension of generalized derivations on modules introduced by Nakajima. Namely, we study homological properties of generalized σ-derivations. Also, we equip the category of functors that send S/R-modules to R-modules of generalized σ-derivations with a tensor product. We show this category is semi-monoidal. As an application, we characterize when a generalized derivation on a path algebra of an acyclic quiver is a generalized inner derivation. [ABSTRACT FROM AUTHOR]

Recall that an R-module M is pure-semisimple if every module in the category σ [ M ] is a direct sum of finitely generated (and indecomposable) modules. A theorem from commutative algebra due to Köthe, Cohen-Kaplansky and Griffith states that "a commutative ring R is pure-semisimple (i.e., every R-module is a direct sum of finitely generated modules) if and only if every R-module is a direct sum of cyclic modules, if and only if, R is an Artinian principal ideal ring". Consequently, every (or finitely generated, cyclic) ideal of R is pure-semisimple if and only if R is an Artinian principal ideal ring. Therefore, a natural question of this sort is "whether the same is true if one only assumes that every proper ideal of R is pure-semisimple?" The goal of this paper is to answer this question. The structure of such rings is completely described as Artinian principal ideal rings or local rings R with the maximal ideals M = R x ⊕ T which Rx is Artinian uniserial and T is semisimple. Also, we give several characterizations for commutative rings whose proper principal (finitely generated) ideals are pure-semisimple. [ABSTRACT FROM AUTHOR]

In this paper, we consider representations of Coxeter groups over a path algebra, R, introduced by Dyer in the study of root systems over non-commutative rings. We answer a question posed by Dyer about the multiplicative properties of R, showing that it is "almost a domain." We also show that R cam be embedded in a matrix ring over a free product of extension fields of the rational numbers and rings of Laurent polynomials. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and let s ≥ 1 be a fixed integer. An additive map h on R is called a homoderivation if h (x y) = h (x) h (y) + h (x) y + x h (y) holds for all x , y ∈ R. In 1978, Herstein proved that a prime ring R of char (R) ≠ 2 is commutative if there is a nonzero derivation d of R such that [ d (x) , d (y) ] = 0 for all x , y ∈ R . The main objective of this paper is to prove the above mentioned result for homoderivations with nilpotency 's' in prime rings. Moreover, we prove that if a prime ring admits homoderivations h1 and h2 such that h 1 ° h 2 = h 2 ° h 1 and h 1 s 1 ° h 2 s 2 = 0 for positive integers s1 and s2, then at least one of the homoderivations must be nilpotent. [ABSTRACT FROM AUTHOR]

This paper centers around one of several generalizations of primary ideals. Which is an intermediate class between S-primary ideals and quasi S-primary ideals. Let R be a commutative ring with identity and S be a multiplicative closed subset of R. A proper ideal I of R disjoint from S is called strongly quasi S-primary if there exists an s ∈ S such that whenever x , y ∈ R and xy ∈ I , then either s x 2 ∈ I or sy ∈ I . Many basic properties of strongly quasi S-primary ideals are given, and examples are presented to distinguish the last concept from other classical ideals. Moreover, forms of strongly quasi S-primary ideals in polynomial rings, power series rings and idealization of a module are investigated. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring and a an ideal of R. The goal of this paper is to establish the local-global principle for the artinianness dimension r a (M) , where r a (M) is the smallest integer such that the local homology module of M is not artinian. For an artinian R-module M with the set Coass R H r a (M) a (M) finite, we show that r a (M) = inf { r a R p ( Hom R (R p , M)) | p ∈ Spec R } . And the class of all modules N such that Coass R N is finite is studied. [ABSTRACT FROM AUTHOR]

This paper establishes a connection between rings, lattices and common meadows. Meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. Common meadows are meadows that introduce, as the inverse of zero, an error term a which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to extend some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective. [ABSTRACT FROM AUTHOR]

This paper introduces a simple graph associated to a commutative ring. Nonzero ideals I and J of a commutative ring R are called Krull dimension-dependent whenever dim R / (I + J) = min { dim R / I , dim R / J } . Based on this, we introduce the dimension graph of R. We study and investigate this graph, and we determine all rings with disconnected dimension graph. Moreover, we introduce strongly Krull dimension-dependent ideals, and we examine these ideals using a subgraph of the dimension graph. [ABSTRACT FROM AUTHOR]

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n (K) (T n (D)) : = { f ∈ T n (K) [ x ] | f (T n (D)) ⊆ T n (D) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n (R 2) (Ω , T n (R 1)) : = { f ∈ T n (R 2) [ x ] | f (Ω) ⊆ T n (R 1) } for some subsets Ω ⊆ T n (R 1) . We generalize Frisch's result and show that Int T n (R 2) (T n (R 1)) : = Int T n (R 2) (T n (R 1) , T n (R 1)) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings. [ABSTRACT FROM AUTHOR]

We show that any n-absorbing ideal must be strongly n-absorbing, which is the first of Anderson and Badawi's three interconnected conjectures on absorbing ideals. We prove this by introducing and studying objects called maximal and semimaximal collections, which are tools for analyzing the multiplicative ideal structure of commutative rings. Furthermore, we discuss and make heavy use of previous work on absorbing ideals done by Anderson and Badawi, Choi and Walker, Laradji, and Donadze, among others. At the end of this paper, we pose three conjectures, each of which implies the last unsolved conjecture made by Anderson and Badawi. [ABSTRACT FROM AUTHOR]

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and M be an R-module. The R-module M is called 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 M to be distributive based on injective envelopes. The proof uses Matlis' results on injective modules. [ABSTRACT FROM AUTHOR]

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However, the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description. [ABSTRACT FROM AUTHOR]

In this article, we define a set of matrices analogous to Vaserstein-type matrices, which were introduced in the paper 'Serre's problem on projective modules over polynomial rings and algebraic K-theory' by Suslin–Vaserstein in 1976. We prove that these are elementary linear matrices. Also, under some conditions, these matrices belong to Petrov's odd unitary group, which is a generalization of all classical groups. We also prove that these matrices generate the Petrov's odd elementary hyperbolic unitary group when the ring is commutative. [ABSTRACT FROM AUTHOR]

A commutative ring R is stable if every non-zero ideal I of R is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the 2-generator property. In the present paper, we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains. [ABSTRACT FROM AUTHOR]

Let A be a commutative ring with identity and A [ [ X 1 , ... , X n ] ] the ring of formal power series in n variables. In this paper, we consider the SFT-property on A and its relation with A [ [ X 1 , ... , X n ] ] . Then, we see when the minimal prime ideals of A [ [ X 1 , ... , X n ] ] are extended from those of A. We discuss the notion of nicely-contractible ideals in A [ [ X 1 , ... , X n ] ] and its relation with the SFT-property. We apply our results to the z 0 − ideals and the z − ideals of A [ [ X 1 , ... , X n ] ] . [ABSTRACT FROM AUTHOR]

In a recent paper, Türkoğlu have studied the rings admitting nonzero homomorphisms between any two non-projective modules (property (T)). His work raises the following question: What are the rings with nonzero maps between non-projective right modules and singular simple modules? As a generalization of the property (T), we consider two families of rings: those rings with nonzero maps from any non-projective module to any singular simple module (property (T1)) and those rings with nonzero maps from any singular simple module to any non-projective module (property (T2)). Complete characterizations of both classes of rings are obtained and it is shown, in particular, that the rings satisfying property (T) are precisely those rings satisfying both properties (T1) and (T2). We show that R satisfies (T) if and only if R has a unique singular simple right R-module, and R is either two-sided Artinian hereditary serial or right completely coretractable. Furthermore, we prove that a commutative ring R has the property (T) if and only if there is a ring decomposition R ≅ A × B , where A is semisimple Artinian and B satisfies one of the following conditions: (1) B is a semi-Artinian max ring with unique singular simple right R-module. (2) B is an Artinian hereditary serial ring with unique singular simple right R-module. [ABSTRACT FROM AUTHOR]

Let D be an integral domain, w be the so-called w-operation on D, and GV (D) be the set of nonzero finitely generated ideals J of D with J − 1 = D . Let I be a w-ideal of D, D/I be the factor ring of D modulo I, E (D / I) be the injective envelope of D/I, and (D / I) w = { x ∈ E (D / I) | J x ⊆ D / I for some J ∈ GV (D) } . Recently Zhou, Kim, and Hu showed that (D / I) w is a commutative ring with identity containing D/I. The Krull domains are a w-analog of the Dedekind domains in the sense that D is a Krull domain if and only if every nonzero ideal of D is w-invertible. It is well known that D is a Dedekind domain if and only if D/aD is a zero-dimensional semilocal principal ideal ring (PIR) for all nonzero nonunits a ∈ D , if and only if D/I is a zero-dimensional semilocal PIR for all nonzero ideals I of D. In this paper, we prove that D is a Krull domain if and only if (D / a D) w is a zero-dimensional semilocal PIR for any nonzero nonunit a of D, if and only if (D / I) w is a zero-dimensional semilocal PIR for any w-ideal I of D. We also study when (D / I) w is Noetherian, Artinian, or a (zero-dimensional) semilocal Bézout ring for all nonzero principal ideals or w-ideals I of D. [ABSTRACT FROM AUTHOR]

Let P be a partially ordered set, R a 2-torsion free commutative ring with unity and F I (P , R) the finitary incidence algebra of P over R. In this paper, we give an explicit description for the structure of nonlinear Jordan derivations of F I (P , R) . We show that if P has no trivial component, then every nonlinear Jordan derivation of F I (P , R) is proper, and can be presented as a sum of a generalized inner derivation, a transitive induced derivation and an additive induced derivation. If P has a trivial connected component, we prove that every nonlinear Jordan derivation of F I (P , R) is proper if and only if every nonlinear Jordan derivation of R is also proper. [ABSTRACT FROM AUTHOR]

This paper investigates when local cohomology modules have an annihilator that does not depend on the choice of an ideal. Takahashi classified the dominant resolving subcategories of the category of finitely generated modules over a commutative Noetherian ring. We show that his classification theorem describes annihilation results of local cohomology modules over a finite-dimensional ring with certain assumptions or a Cohen-Macaulay ring. [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]

Let R be a commutative ring and Nil(R) denote the set of nilpotent elements of R. In this paper, we investigate two generalizations of exact sequences, ϕ -exact sequences and N -exact sequences. With the help of two generalizations of exact sequences, we generalize the concept of homology modules. We show that they behave in a way similar to the classical ones. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. An R-module M is said to be w-split if Ext R 1 (M , N) is a GV-torsion R-module for all R-modules N. It is known that every projective module is w-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) projective modules, which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An R-module M is said to be w-copure projective (resp., strongly w-copure projective) if Ext R 1 (M , N) (resp., Ext R n (M , N) ) is a GV-torsion R-module for all flat R-modules N and any n ≥ 1 . [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of ring called regular S-Noetherian ring, which is a weak version of S-Noetherian ring property. Any S-Noetherian ring is naturally a regular S-Noetherian ring, and in the domain context, these two forms coincide. We study the transfer of this notion to various context of commutative ring extensions such as direct product, trivial ring extensions and amalgamated duplication of a ring along an ideal. Our results generate new families of examples of non-S-Noetherian regular S-Noetherian rings. [ABSTRACT FROM AUTHOR]

In this paper, we determine the energies of complements of unitary one-matching bi-Cayley graphs over finite commutative rings. We also determine the energies of some one-matching bi-gcd-graphs as well as the energies of complements of some one-matching bi-gcd-graphs, which are generalizations of unitary one-matching bi-Cayley graphs. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. We say that R has the unique decomposition into regular ideals (UDRI) property if, for any R-module which decomposes into a finite direct sum of regular ideals, this decomposition is unique up to the order and isomorphism class of the regular ideals. In this paper, we will prove some preliminary results for Marot rings whose regular ideals are finitely generated and give a necessary and sufficient condition for these rings to satisfy the UDRI property. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph in which the vertex set is R, and two distinct vertices x and y are adjacent if and only if x + y is a unit in R. In this paper, we determine when these graphs are well-covered, and then, by applying this result, we characterize the unit graphs whose edge rings are Cohen–Macaulay (Gorenstein). This characterization gives us a large class of non-Cohen–Macaulay graphs. [ABSTRACT FROM AUTHOR]

Recently, in a series of papers "simple" versions of direct-injective and direct-projective modules have been investigated. These modules are termed as "simple-direct-injective" and "simple-direct-projective," respectively. In this paper, we give a complete characterization of the aforementioned modules over the ring of integers and over semilocal rings. The ring is semilocal if and only if every right module with zero Jacobson radical is simple-direct-projective. The rings whose simple-direct-injective right modules are simple-direct-projective are fully characterized. These are exactly the left perfect right H-rings. The rings whose simple-direct-projective right modules are simple-direct-injective are right max-rings. For a commutative Noetherian ring, we prove that simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. Various closure properties and some classes of modules that are simple-direct-injective (resp. projective) are given. [ABSTRACT FROM AUTHOR]

Let S be a (multiplicative) commutative semigroup with 0, Z(S) the set of zero-divisors of S, and n a positive integer. The zero-divisor graph of S is the (simple) graph Γ (S) with vertices Z (S) * = Z (S) ∖ { 0 } , and distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we introduce and study the n-zero-divisor graph of S as the (simple) graph Γ n (S) with vertices Z n (S) * = { x n | x ∈ Z (S) } ∖ { 0 } , and distinct vertices x and y are adjacent if and only if xy = 0. Thus each Γ n (S) is an induced subgraph of Γ (S) = Γ 1 (S). We pay particular attention to diam (Γ n (S)) , g r (Γ n (S)) , and the case when S is a commutative ring with 1 ≠ 0. We also consider several other types of "n-zero-divisor" graphs and commutative rings such that some power of every element (or zero-divisor) is idempotent. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative B C -linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand–Mazur theorem for the bicomplex division Banach algebra. [ABSTRACT FROM AUTHOR]

Let R be a 2-torsion free commutative ring with unity and X be a locally finite preordered set. We prove in this paper that every nonlinear Jordan higher derivation on the incidence algebra I (X , R) is an additive higher derivation, provided that all connected components of X are nontrivial. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with non-zero identity. Let R × and J (R) be the group of unit elements and the Jacobson radical of R, respectively. The unit graph of the ring R, denoted by G (R) , is a graph whose vertex set is R and two distinct vertices x and y are adjacent if and only if x + y ∈ R × . If we relax this definition by dropping the term 'distinct', we obtain the closed unit graph, denoted by G ¯ (R). In this paper, we compute the adjacency spectrum of the graph G ¯ (R). We utilize this result to show that G (R) ≅ G (S) if and only if (R / J (R)) ≅ (S / J (S)) and | J (R) | = | J (S) | , where R and S are two arbitrary finite rings. Moreover, we determine when G (R) is a Ramanujan graph. We also deliver a necessary and sufficient condition for G (R) to be a strongly regular graph. Finally, we obtain the spectrum of a generalization of both unit and unitary Cayley graphs. [ABSTRACT FROM AUTHOR]

The paper contains a simple solution of Stratton and Webb's problem related to the so-called absolute annihilators of abelian groups researched also by Feigelstock. Namely, there are provided constructions of indecomposable torsion-free abelian groups G of any finite rank greater than two for which the quotient group modulo the subgroup generated by squares of all possible rings on G supports a non-zero associative ring structure. They are much less complicated than those obtained before and show that groups satisfying the mentioned condition are not as rare phenomena as might be supposed. Moreover, most of these constructions led to new examples of CR- and AR-groups, i.e. abelian groups supporting only commutative and only associative rings, respectively. In particular, the first examples of indecomposable as well as decomposable AR-groups which do not satisfy the condition CR are presented. The first example of an indecomposable torsion-free group satisfying the condition CR only in the class of associative rings is also obtained. [ABSTRACT FROM AUTHOR]

This paper is a new contribution to the partial Galois theory of groups. First, given a unital partial action αG of a finite group G on an algebra S such that S is an α G -partial Galois extension of S α G and a normal subgroup H of G, we prove that α G induces a unital partial action α G / H of G/H on the subalgebra of invariants S α H of S such that S α H is an α G / H -partial Galois extension of S α G . Second, assuming that G is abelian, we construct a commutative inverse semigroup T par (G , R) , whose elements are equivalence classes of α G -partial abelian extensions of a commutative algebra R. We also prove that there exists a group isomorphism between T par (G , R) / ρ and T(G, A), where ρ is a congruence on T par (G , R) and T(G, A) is the classical Harrison group of the G-isomorphism classes of the abelian extensions of a commutative algebra A. It is shown that the study of T par (G , R) reduces to the case where G is cyclic. The set of idempotents of T par (G , R) is also investigated. [ABSTRACT FROM AUTHOR]

Let R be non-commutative ring with unity and n ≥ p ≥ 2 , p prime. In this paper, we prove that an n × n matrix over R is the sum of pth powers if and only if its trace can be written as a sum of pth powers and commutators modulo pR. This extends the results of L. N. Vaserstein (p = 2) and S. A. Katre, Kshipra Wadikar (p = 3). We also obtain necessary and sufficient conditions for a matrix over R to be written as a sum of fourth powers when n ≥ 2. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, R∙ be the multiplicative monoid of regular elements in R, t be the so-called t-operation on R or R∙. A Marot ring is a ring whose regular ideals are generated by their regular elements. Marot rings were introduced by J. Marot in 1969 and have been playing a key role in the study of rings with zero divisors. The notion of Marot rings can be extended to t-Marot rings such that Marot rings are t-Marot rings. In this paper, we study some ideal-theoretic relationships between a t-Marot ring R and the monoid R∙. We first construct an example of a t-Marot ring that is not Marot. This also serves as an example of a rank-one DVR of reg-dimension ≥2. Let R be a t-Marot ring, t-spec(R) (resp., t-spec(R.)) be the set of regular prime t-ideals of R (resp., the set of non-empty prime t-ideals of R∙), and Cl(A) be the class group of A for A = R or R∙. Then, among other things, we prove that the map φ:t-spec(R)→t-spec(R∙) given by φ(P)=P∙ is bijective; Cl(R)≅Cl(R∙); and R is a factorial ring if and only if R∙ is a factorial monoid.. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring of characteristic and G be a group. It is known that the group ring RG is bounded Lie Engel if and only if either G is nilpotent and G has a p-Abelian normal subgroup of finite p-power index (if n is a power of a prime p) or G is Abelian. In this paper we try to generalize this result; if x and y are elements of RG, let [x,y] = xy – yx and inductively,. Let m and n be two natural numbers. Among other results, we show that if RG satisfies , then either is a p-group (if n is a power of a prime p) or G is Abelian. If G is locally finite, then we show that RG satisfies if and only if either G is locally nilpotent and is a p-group (if n is a power of a prime p) or G is Abelian. [ABSTRACT FROM AUTHOR]