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There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics. [ABSTRACT FROM AUTHOR]

In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into optical art by depending on the reflection of vertices, edges, and planes on the x or y-axes. That is, we achieve Op art solely through pure mathematics in this paper. [ABSTRACT FROM AUTHOR]

We fill a gap in the proof of one of the central theorems in Epp’s paper, concerning p-cyclic extensions of complete discrete valuation rings. [ABSTRACT FROM AUTHOR]

This paper presents an algorithm for calculation of the Lyubeznik numbers of a local ring which is a homomorphic image of a regular local ring R of prime characteristic. The methods used employ Lyubeznik's F-modules over R, particularly his F-finite F-modules, and also the modules of generalized fractions of Sharp and Zakeri [Mathematika 29 (1982), pp. 32–41]. It is shown that many modules of generalized fractions over R have natural structures as F-modules; these lead to F-module structures on certain local cohomology modules over R, which are exploited, in conjunction with F-module structures on injective R-modules that result from work of Huneke and Sharp [Trans. Amer. Math. Soc. 339 (1993), pp. 765–779], to compute Lyubeznik numbers. The resulting algorithm has been implemented in Macaulay2. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if a R + b R = e R for some nonzero idempotent e in R. Let G ′ (R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G ′ (R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G ′ (R) is ≤ 2 . Finally, we show that the graph G ′ (R) is a line graph of some graph if and only if R is either a regular ring or a local ring. [ABSTRACT FROM AUTHOR]

Let τ be a finite type hereditary torsion theory on the category of all modules over a commutative ring. The purpose of this paper is to give two applications of Nagata rings and modules in the sense of Jara [Nagata rings, Front. Math. China10 (2015) 91–110]. First they are used to obtain Chase's Theorem for τ -coherent rings. In particular, we obtain the w -version of Chase's Theorem, where w is the classical star operation in ideal theory. In the second half, we apply they to characterize τ -flatness in the sense of Van Oystaeyen and Verschoren [Relative Invariants of Rings-The Commutative Theory, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 79 (Marcel Dekker, Inc., New York, 1983)]. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and I a fixed ideal R of and M be an unitary R-module. In this paper we introduce and study the concept of I-nearly prime submodules as genrealizations of nearly prime and we investigate some properties of this class of submodules. Also, some characterizations of I-nearly prime submodules will be given. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with nonzero identity and n be a positive integer. In this paper, we study the concepts of n-absorbing δ-primary ideals and weakly n-absorbing δ-primary ideals, which are the generalizations of δ-primary ideals and weakly δ-primary ideals, respectively. We introduce the concepts of n-absorbing δ-primary ideals and weakly n-absorbing δ-primary ideals. Moreover, we give many properties of these new types of ideals and investigate the relations between these structures. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and M be an R-module. In this paper some properties of primary and classical primary submodules will be investigated. It is shown that if F is a faithfully flat R-module, then Q is a primary submodule of M if and only if F ⊗ Q is a primary submodule of F ⊗ M. Also some characterization of classical primary submodules are given. Furthermore, the existence of (minimal) primary submodules containing classical primary submodules are investigated. [ABSTRACT FROM AUTHOR]

In this paper, we study on modules that have a weak (ample) δ-supplement in every extension which are adapted Zöschinger's modules with the properties (E) and (EE). It is shown that: (1) Direct summands of modules with the property δ-(CWE) have the property δ-(CWE); (2) For a module M, if every submodule of M has the property δ-(CWE) then so does M; (3) For a ring R, R is δ-semilocal iff every R-module has the property δ-(CWE); (4) Every factor module of a finitely generated module that has the property δ-(CWE) also has the property δ-(CWE) under a special condition; (5) Let M be a module and L be a submodule of M such that L <<#948; M. If the factor module M/L has the property δ-(CWE), then so does M; (6) On a semisimple module the concepts of modules that have the property δ-(CE) and δ-(CWE) coincide with each other. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study generalized derivations in rings with involution which behave like strong commutativity preserving mappings. In fact, we prove the following result: Let R be a noncommutative prime ring with involution of the second kind such that char . If R admits a generalized derivation associated with a derivation such that for all , then for all or for all . Moreover, a related result is also obtained. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the bases of a semimodule over a commutative semiring R are investigated. Some properties and characterizations of the bases are discussed and some equivalent conditions for a basis to be a free basis in a finitely generated free semimodule over R are given. The different possible cardinalities for a basis in a finitely generated free semimodule over R are considered and some equivalent descriptions are obtained for a commutative semiring R satisfying the property that any two bases for a finitely generated free R-semimodule have the same cardinality. Partial results obtained in the paper develop and generalize the corresponding results for commutative join-semirings and for commutative zerosumfree semirings. [Copyright &y& Elsevier]

The paper adjoins the work of B. I. Plotkin and S. M. Vovsi, Varieties of Representations of Groups (Zinatne, Riga (1983)), and turns out to be, in a sense, its continuation. In the former, the varieties of representations have been considered. As a matter of fact, the varieties under consideration are action-type varieties. This paper studies other classes of representations, axiomatizable in a special action-type logic. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary finite commutative local ring. In this paper, we obtain a necessary and sufficient condition for a function over R to be a polynomial function. Before this paper, necessary and sufficient conditions for a function to be a polynomial function over some special finite commutative local rings were obtained. [ABSTRACT FROM AUTHOR]

Abstract: This paper is motivated by Ito''s paper ((Hokkaido) Math. J. 22 (1993) 373), in which he constructed -relative difference sets in . Chen and Li have shown that , the alternating group of degree 5, has a (30,2,29,14)-relative difference set (see Discrete Math. 256 (2002) 301). These are the only examples of nontrivial relative difference sets in nonsolvable groups known to the author. In this paper, we study relative difference sets in and show that there exist exactly two nontrivial relative difference sets in up to equivalence with the aid of a computer. [Copyright &y& Elsevier]

In this paper all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical primary sub- module, if for each m ∈ M and elements a, b ∈ R, abm ∈ N implies that either am ∈ N or bt m ∈ N for some t ≥ 1. We introduce the notion of "weakly classical primary submodules". A proper submodule N of M is a weakly classical primary submodule if whenever a, b ∈ R and m ∈ M with 0 ≠ abm ∈ N, then either am ∈ N or b m ∈ N for some t ≥ 1. [ABSTRACT FROM AUTHOR]

In this paper, we further study Tate cohomology of modules over a commutative ring with respect to semidualizing modules using the ideals of Sather-Wagstaff et al. [‘Tate cohomology with respect to semidualizing modules’, J. Algebra 324 (2010), 2336–2368]. In particular, we prove a balance result for the Tate cohomology ${\widehat{\mathrm{Ext} }}^{n} $ for any $n\in \mathbb{Z} $. This result complements the work of Sather-Wagstaff et al., who proved that the result holds for any $n\geq 1$. We also discuss some vanishing properties of Tate cohomology. [ABSTRACT FROM PUBLISHER]

The object of this paper is to introduce a permuting tri-derivation in a Γ-near-ring. We obtain the conditions for a prime Γ-near-ring to be a commutative Ν-ring. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and let n > 1 be an integer. A proper submodule N of an R-module M will be called 2-absorbing [resp. n-weakly prime], if r; s ∊ R and x ∊M with rsx ∊ N [resp. rsx ∊ N\(N :M)n-1N] implies that rs ∊ (N :M) or rx ∊ N, or sx ∊ N: These concepts are generalizations of the notions of 2-absorbing ideals and weakly prime submodules, which have been studied in [3, 4, 6, 7]. We will study 2-absorbing and n- weakly prime submodules in this paper. Among other results, it is proved that if (N:M)n-1N ≠ (N:M)2N; then N is 2-absorbing if and only if it is n-weakly prime. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, nilpotent subsemigroups in the matrix semigroup over a commutative antiring are discussed. Some basic properties and characterizations for the nilpotent subsemigroups are given, and some equivalent conditions for the matrix semigroup over a commutative antiring to have a maximal nilpotent subsemigroup are obtained. Also, the maximal nilpotent subsemigroups in the matrix semigroup are described. [Copyright &y& Elsevier]

Abstract: A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, “a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring”. Therefore, an interesting natural question of this sort is “whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?” The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. [Copyright &y& Elsevier]

In this paper we introduce a stronger type of primary decomposition of a Noetherian ring. We call such a ring a Transparent ring and show that if R is a commutative Noetherian ring, which is also an algebra over Q (the field of rational numbers); σ an automorphism of R and δ a σ-derivation of R such that δ(σ(a)) = σ(δ(a)), for all a ϵ R. Further more if aσ(a) ϵ P(R) implies that a ϵ P(R), (P(R) the prime radical of R), then R[χ;σ,δ] is a Transparent ring. [ABSTRACT FROM AUTHOR]

Let R be an associative ring, λ a nonzero left ideal of R, d : R → R a derivation and G : R → R a generalized derivation. In this paper, we study the following situations in prime and semiprime rings: (1) G (x o y) = a(xy ± yx); (2) G[x, y] = a(xy ± yx); (3) d(x) o d(y) = (xy ± yx); for all x, y ∈ λ and a ∈ {0, 1, -1}. [ABSTRACT FROM AUTHOR]

Abstract: An antiring is a semiring which is zerosumfree (i.e., implies for any in this semiring). In this paper, we study the nilpotency of matrices over commutative antirings. We first provide some properties and characterizations of the nilpotent matrices in terms of principal permanental minors, main diagonals and permanental adjoint matrices. When a family of matrices are simultaneously considered, we establish some characterizations of the simultaneous nilpotence for a family of matrices. [Copyright &y& Elsevier]

Abstract: Let be a commutative ring. The total graph of , denoted by is a graph with all elements of as vertices, and two distinct vertices , are adjacent if and only if , where denotes the set of zero-divisors of . Let regular graph of , , be the induced subgraph of on the regular elements of . Let be a commutative Noetherian ring and is not an ideal. In this paper we show that if is a connected graph, then . Also, we prove that if is a finite ring, then is a Hamiltonian graph. Finally, we show that if is a commutative Noetherian ring and is finite, then is finite. [Copyright &y& Elsevier]

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented. [ABSTRACT FROM AUTHOR]

The Nakayama conjecture is one of the most important conjectures in ring theory. The Auslander-Reiten conjecture is closely related to it. The purpose of this paper is to show that if the Auslander-Reiten conjecture holds in codimension one for a commutative Gorenstein ring $R$, then it holds for $R$. [ABSTRACT FROM AUTHOR]

Abstract.  In this paper we establish several equivalent conditions for every regular element (quasi-regular ideal) of a principally generated C-lattice L (a commutative ring with identity R) to be an invertible element of L (quasi-invertible ideal of R). Using these results, some new characterizations are given for almost Dedekind lattices, almost Dedekind rings, Dedekind lattices and Dedekind rings. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we offer a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniform treatment of a number of standard results on prime ideals in commutative algebra, due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, and others. More significantly, the simple nature of this Prime Ideal Principle enables us to generate a large number of hitherto unknown results of the “maximal implies prime” variety. The key notions used in our uniform approach to such prime ideal problems are those of Oka families and Ako families of ideals in a commutative ring, defined in (2.1) and (2.2). Much of this work has also natural interpretations in terms of categories of cyclic modules. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study families of Artinian or one-dimensional quotients of a polynomial ring R with a special look to level algebras. Let be the scheme parametrizing graded quotients of R with Hilbert function H. Let be any graded surjection of quotients of R with Hilbert function and , respectively. If (respectively ) and A is a “truncation” of B in the sense that (respectively ) for some , then we show there is a close relationship between and concerning e.g. smoothness and dimension at the points and , respectively, provided B is a complete intersection or provided the Castelnuovo–Mumford regularity of A is at least 3 (sometimes 2) larger than the regularity of B. In the complete intersection case we generalize this relationship to “non-truncated” Artinian algebras A which are compressed or close to being compressed. For more general Artinian algebras we describe the dual of the tangent and obstruction space of graded deformations in a manageable form which we make rather explicit for level algebras of Cohen–Macaulay type 2. This description and a linkage theorem for families allow us to prove a conjecture of Iarrobino on the existence of at least two irreducible components of , , whose general elements are Artinian level algebras of type 2. [Copyright &y& Elsevier]

Let R be a ring with center Z(R). The aim of this paper is to investigate the commutativity of R satisfying any one of the following properties: (i) F(xy) xy Ε Z(R), (ii) F(xy) + xy Ε Z(R), (iii) F(xy) - yx ε Z(R), (iv) F(xy) + yx ε Z(R), (v) F(x)F(y) - xy ε Z(R) and (vi) F(x)F(y) + xy ε Z(R), for all x, y ε R, where F is a generalized derivation on R. [ABSTRACT FROM AUTHOR]

Abstract: Let R be an arbitrary commutative ring with identity, gl(n, R) the general linear Lie algebra over R. The aim of this paper is to give an explicit description of the derivation algebras of the parabolic subalgebras of gl(n, R). [Copyright &y& Elsevier]

In this paper, we introduce a notion of weak Fountain-Gould left order for associative pairs and give a Goldie-like theory of associative pairs which are weak Fountain-Gould left orders in semiprime pairs coinciding with their socles. [ABSTRACT FROM AUTHOR]

In this paper, we investigate and characterize the class of A-semirings. A characterization of the Thierrin radical of a proper ideal of an A-semiring is given. Moreover, when P is a Q-ideal in the semiring R, it is shown that P is primary if and only if R/P is nilpotent. [ABSTRACT FROM AUTHOR]

Let v be a henselian valuation of a field K, and $\bar{v}$ be the (unique) extension of v to a fixed algebraic closure $\overline{K}$ of K. For an element $\theta\in \overline{K} \backslash K$, a chain $\theta=\theta_{0},\theta_{1},\dots, \theta_{m}$ of elements of $\overline {K}$ such that θi is of minimal degree over K with the property that $\bar{v}(\theta_{i-1}-\theta_{i}) =\sup\{\bar{v}(\theta_{i-1}-\beta) \mid [K(\beta):K] < [K(\theta_{i-1}):K]\}$ and θm ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each $\theta \in \overline{K} \backslash K$ when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements $\theta\in \overline {K} \backslash K$ for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ. [ABSTRACT FROM AUTHOR]

Abstract: This paper gives criteria for a Cohen–Macaulay local ring to be Gorenstein, in terms of the vanishing of Ext modules. The main results extend earlier work of Ulrich. [Copyright &y& Elsevier]

Let R be an associative ring. An additive mapping F : R → R is called a generalized derivation if there exists a derivation d : R → R such that F(xy) = F(x)y + xd(y) holds for all x, y ∈ R. In this paper, we explore the commutativity of the ring R satisfying any one of the following conditions : (i) d(x) ∘ F(y) = 0, (ii) [d(x),F(y)] = 0, (iii) d(x) ∘ F(y) = x ∘ y, (iv) d(x) ∘ F(y) + x ∘ y = 0, (v) d(x)F(y) - xy ∈ Z(R), (vi) d(x)F(y) + xy ∈ Z(R), (vii) [d(x), F(y)] = [x, y] and (viii) [d(x), F(y)] + [x, y] = 0, for all x, y in some appropriate subset of the ring R. [ABSTRACT FROM AUTHOR]

A right ideal I of a ring R is said to be a comparizer right ideal of R if for any right ideals A , B of R , either A ? B or BI ? A . A ring R (with 1) is called a right pseudo-chain ring if for any nonunit a ? R , aR is a comparizer right ideal of R . These rings are generalizations of right chain rings and commutative pseudo-valuation rings. In this paper we study the structure of right pseudo-chain rings. We prove that a ring R containing a non-zero-divisor in the Jacobson radical J ( R ) is a pseudo-chain ring if and only if R is a local subring of a (unique up to isomorphism) chain ring T with J ( T ) = J ( R ). We also introduce the concept of a pseudo-uniserial module. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce the concept of integrally closed modules and characterize Dedekind modules and Dedekind domains. They also show that a given domain R is integrally closed if and only if a finitely generated torsion-free projective R -module is integrally closed. In addition, it is proved that any invertible submodule of a finitely generated projective module over a domain is finitely generated and projective. Also they give the equivalent conditions for Dedekind modules and Dedekind domains. [ABSTRACT FROM AUTHOR]

Let R be a differential domain finitely generated over a differential field F of characteristic 0. Let C be the subfield of differential constants of F . This paper investigates conditions on differential ideals of R that are necessary or sufficient to guarantee that C is also the set of constants of differentiation of the quotient field, E , of R . In particular, when C is algebraically closed and R has a finite number of height one differential prime ideals, there are no new constants in E . An example where F is infinitely generated over C shows the converse is false. If F is finitely generated over C and R is a polynomial ring over F , sufficient conditions on F are given so that no new constants in E does imply only finitely many height one prime differential ideals in R . In particular, F can be Q ( T ) where T is a finite transcendence set. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism s ? Aut( R ), we use the notion of a s-compatible module M R to obtain results on the couniform dimension of the polynomial modules M [ x ], M [ x -1 ], and M [ x , x -1 ] over suitable skew extension rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and let N be a non-zero R-module. Let φ be a multiplicatively cloud set of non-zero ideals of R. For a submodule M of N, we define the φ-closure of M as Mφ - ∪I∈φ(M :N I). The purpose of this paper is to show that the operation M → Mφ is a closure operation on the set of submodules M of N, and if N satisfies the ACC on colon submodules, then it is a semi-prime operation that satisfies the φ-cancellation law. Further, N has a natural φ-closure Nφ, and A ⊗R N → (A ⊗R N)φA is a closure operation on any set M = {A ⊗R N|A is a ring, R ⊆ A and if C is a ring between R and A, then C ⊗R N ⊆ A ⊗R N and C\Zc(C ⊗R N) ⊆ A\ZA(A ⊗R N)}. Finally, if N is Noetherian, then the associated primes of Mφ are also associated primes of IMφ, (IM)φ and IφMφ for all I ∈ φ. [ABSTRACT FROM AUTHOR]

In this paper we study dual coalgebras of algebras over arbitrary (Noetherian) commutative rings. We present and study a generalized notion of coreflexive comodules and use the results obtained for them to characterize the so called coreflexive coalgebras. Our approach in this note is an algebraically topological one. [Copyright &y& Elsevier]

Let (R,m) be a Noetherian local ring of prime characteristic p. In this paper, as an extension of the concept of pn-basis, we introduce the notion of m-adic pn-basis, and we show that R/Rpn has an m-adic pn-basis for every n (n=1,2,…) if and only if R is a regular local ring. For any Noetherian local ring, further we introduce the concept of m-adic free graded algebra and we give a regularity criterion in terms of the higher differential algebra, and others. [Copyright &y& Elsevier]

Let G be a group and R be a G-graded commutative ring, i.e., ... and Rg Rh∈ Rgh for all g, h ⊆ G. In this paper, we study the graded primary ideals and graded primary G-decomposition of a graded ideal. [ABSTRACT FROM AUTHOR]

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting. [ABSTRACT FROM AUTHOR]