Search

Omega rings (Ω-rings) and some special types of these structures (commutative, without zero divisors, Ω-fields,...) are introduced and investigated. Ω is a complete lattice and Ω-rings are lattice-valued algebraic structures, defined on crisp algebras of the same type as rings, but which are not necessarily rings. The classical equality is replaced by a lattice-valued one. We give some basic properties of Ω-rings and related structures. [ABSTRACT FROM AUTHOR]

We prove a necessary condition and a sufficient condition for an n × n -matrix A with determinantal rank ρ (A) = t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its t th compound matrix C t (A). In particular, a non-zero n × n -matrix A with ρ (A) = t over a local commutative ring R is strongly regular if and only if Tr (C t (A)) is a unit in R , and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras. [ABSTRACT FROM AUTHOR]

The multiplication table of a finite field F q produces a difference matrix over the elementary abelian group (F q , +). This result was extended to a ring by Jungnickel (1979), which yields a difference scheme of strength 2. This work deals with constructions of difference schemes of strength t over a finite commutative ring with identity for an arbitrary t ≥ 2 , generalizing the previous well-known result. As consequences, we generalize a result on t -simple matrix by Chu and Golomb (2003) and a result on difference schemes by Hedayat, Sloane and Stufken (2012) by constructing difference schemes over a Galois ring. This work also discusses some applications to orthogonal arrays and covering scheme numbers. In particular, the results enable us to derive a classical result by Bush (1952) on orthogonal arrays. [ABSTRACT FROM AUTHOR]

If R is a commutative unital ring and M is a unital R -module, then each element of End R (M) determines a left End R (M) [ X ] -module structure on End R (M) , where End R (M) is the R -algebra of endomorphisms of M and End R (M) [ X ] = End R (M) ⊗ R R [ X ]. These structures provide a very short proof of the Cayley-Hamilton theorem, which may be viewed as a reformulation of the proof in Algebra by Serge Lang. Some generalisations of the Cayley-Hamilton theorem can be easily proved using the proposed method. [ABSTRACT FROM AUTHOR]

We study an analogue of the Collatz map in the polynomial ring R [ x ] , where R is an arbitrary commutative ring. We prove that if R is of positive characteristic, then every polynomial in R [ x ] is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on F p [ x ] and F 2 [ x ] , respectively. We also consider the Collatz map on the ring of formal power series R [ [ x ] ] when R is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring Z 2 of 2-adic integers, extending previous results of Lagarias. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. We determine the cyclic modules in the projectivity domain and those in the injectivity domain of a simple R -module. An R -module M is called poor (p-poor) if for every R -module N , M is N -injective (M is N -projective) only if N is semisimple. We describe the structure of simple poor R -modules and simple p-poor R -modules. We show that every simple poor R -module is p-poor and the converse holds when R is noetherian or semilocal. It is shown that every simple R -module is poor if and only if every simple R -module is p-poor if and only if R is local or semisimple. We prove that every simple R -module is either p-poor or projective if and only if R is semisimple or R has exactly one essential maximal ideal. We conclude by studying the class of commutative rings R over which every simple R -module is either poor or injective. [ABSTRACT FROM AUTHOR]

Let R be an arbitrary finite commutative ring with unity. The zero-divisor graph of R , denoted by Γ (R) , is a graph with vertex set non-zero zero-divisors of R and two of them are connected by an edge if their product is zero. In this paper, we derive a formula for the Wiener index of the graph Γ (R). In the literature, the Wiener index of the graph Γ (R) is known only for R = Z n , the ring of integers modulo n. As applications of our formula, the Wiener index of Γ (R) is explicitly calculated when (i) R is a reduced ring, (ii) R is the ring of integers modulo n , and (iii) more generally R is the product of ring of integers modulo n. The Wiener index of the zero-divisor graph of the ring of Gaussian integers over Z n is also discussed. [ABSTRACT FROM AUTHOR]

Recursive MDS matrices are used for the design of linear diffusion layers in lightweight cryptographic applications. Most of the works on the construction of recursive MDS matrices either consider matrices over finite fields or block matrices over G L (m , F 2). In the first case, there have been works on the direct construction of recursive MDS matrices. The latter case is hard to deal with because of its non-commutative nature. There has not been any serious attempt to look for recursive MDS matrices over finite commutative rings, in particular over local rings of even characteristic. In this work, we present several methods for the construction of recursive MDS companion matrices over finite commutative rings. The main tools are the simple expressions for the determinant of (generalized) Vandermonde and linearized matrices. We show that the determinant of a linearized matrix over a finite commutative ring of prime characteristic can be expressed in a simple form. We discuss a technique called subring construction with which MDS matrices over product rings can be constructed using MDS matrices over subrings. We give a few examples of recursive MDS companion matrices over local rings of even characteristic. We also discuss some results on the nonexistence of recursive MDS matrices over certain rings for some parameter choices. [ABSTRACT FROM AUTHOR]

According to the algebraic approach to spacetime, a thoroughgoing dynamicism, physical fields exist without an underlying manifold. This view is usually implemented by postulating an algebraic structure (e.g., commutative ring) of scalar-valued functions, which can be interpreted as representing a scalar field, and deriving other structures from it. In this work, we point out that this leads to the unjustified primacy of an undetermined scalar field. Instead, we propose to consider algebraic structures in which all (and only) physical fields are primitive. We explain how the theory of natural operations in differential geometry—the modern formalism behind classifying diffeomorphism-invariant constructions—can be used to obtain concrete implementations of this idea for any given collection of fields. For concrete examples, we illustrate how our approach applies to a number of particular physical fields, including electrodynamics coupled to a Weyl spinor. [ABSTRACT FROM AUTHOR]

We show that there is an effectively given zero-sum-free commutative semiring S , contained as the subsemiring of nonnegative elements in an effectively given commutative ordered ring, for which there are no procedures deciding, given a weighted finite automaton over S , whether its support is the language of all words or whether its support is infinite. In particular, by a result of D. Kirsten (2011), since S is zero-sum-free and commutative, the support is recognizable by a classical finite automaton, but such an automaton or even just a pushdown automaton for its support cannot be constructed effectively. • We construct an effectively given zero-sum-free commutative semiring S as a subsemiring of a commutative ordered ring. • It is undecidable, given a weighted finite automaton over S , whether its support is the language of all words. • It is also undecidable whether the support is infinite. • By a result of D. Kirsten (2011), the support is recognizable by a classical finite automaton. • But such an automaton or even just a pushdown automaton for its support cannot be constructed effectively. [ABSTRACT FROM AUTHOR]

There is a natural connection between the third homology of SL 2 (A) and the refined Bloch group RB (A) of a commutative ring A. In this article we investigate this connection and as the main result we show that if A is a universal GE 2 -domain such that − 1 ∈ A × 2 , then we have the exact sequence H 3 (SM 2 (A) , Z) → H 3 (SL 2 (A) , Z) → RB (A) → 0 , where SM 2 (A) is the group of monomial matrices in SL 2 (A). Moreover, we show that RP 1 (A) , the refined scissors congruence group of A , is naturally isomorphic to the relative homology group H 3 (SL 2 (A) , SM 2 (A) ; Z). [ABSTRACT FROM AUTHOR]

A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original Caldero-Chapoton map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and Jørgensen constructed a modified Caldero-Chapoton map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient K 0 sp (T) / M of a Grothendieck group of a cluster tilting subcategory T is used. In this article, we show that this quotient is the Grothendieck group of a certain extriangulated category, thereby exposing the significance of it and the relevance of extriangulated structures. We use this to define another modified Caldero-Chapoton map that recovers the one of Holm–Jørgensen. We prove our results in a higher homological context. Suppose S is a (d + 2) -angulated category with subcategories X ⊆ T ⊆ S , where X is functorially finite and T is 2 d -cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group K 0 (S , E X , s X) of the category S , equipped with the d -exangulated structure induced by X , and the quotient K 0 sp (T) / N , where N is the higher analogue of M above. When X = T the isomorphism is induced by the higher index with respect to T introduced recently by Jørgensen. Thus, in the general case, we can understand the map taking an object in S to its K 0 -class in K 0 (S , E X , s X) as a higher index with respect to the rigid subcategory X. [ABSTRACT FROM AUTHOR]

Let A be a complete and cocomplete abelian category with the additional assumption that any direct sum and product of short exact sequences are exact. We explore the relation between balanced pairs and TTF triples in A. The main results are: (1) every balanced pair in A induces a TTF triple; (2) if A has projective covers and injective envelopes, then every TTF triple in A gives rise to a balanced pair, and hence there is a bijection between the equivalence classes of balanced pairs and TTF triples in A ; (3) a balanced pair in A is quasi admissible if and only if its induced TTF triple is centrally splitting. Our first application of these results provide abundant rings over which every balanced pair is quasi admissible, including local rings, commutative semiperfect rings, and commutative Noetherian rings. Another application is the classification of equivalence classes of cohereditary balanced pairs over arbitrary rings. We also present counterexamples to [15, Open questions 3 and 5]. Finally, we prove that the answers to [15, Open questions 2 and 4] are positive for coherent rings with weak global dimension at most one. [ABSTRACT FROM AUTHOR]

Given a non-necessarily commutative unital ring R and a unital partial representation Θ of a group G into the Picard semigroup PicS (R) of the isomorphism classes of partially invertible R -bimodules, we construct an abelian group C (Θ / R) formed by the isomorphism classes of partial generalized crossed products related to Θ and identify an appropriate second partial cohomology group of G with a naturally defined subgroup C 0 (Θ / R) of C (Θ / R). Then we use the obtained results to give an analogue of the Chase-Harrison-Rosenberg exact sequence associated with an extension of non-necessarily commutative rings R ⊆ S with the same unity and a unital partial representation G → S R (S) of an arbitrary group G into the monoid S R (S) of the R -subbimodules of S. This generalizes the works by Kanzaki and Miyashita. [ABSTRACT FROM AUTHOR]

Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of ℓ -dimension linear intersection pairs (ℓ -DLIPs) of codes over a finite commutative ring (R), for some positive integer ℓ. In this paper, we study ℓ -DLIP of codes over R in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) ℓ -DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring which helps us to get an important characterization of ℓ -DLIP of constacyclic codes. Finally, the ℓ -DLIP of constacyclic codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes. [ABSTRACT FROM AUTHOR]

Given a representation of a finite group G over some commutative base ring k , the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k -algebra automorphisms, then the cofixed space is a module over the ring of G -invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p. [ABSTRACT FROM AUTHOR]

We establish a duality between injective envelopes and flat covers over a commutative Noetherian ring. One case of this duality states that a morphism is an injective envelope, if and only if its Matlis dual is a flat cover. We also show that if we swap injective envelopes and flat covers in this duality, neither implication is true in general. [ABSTRACT FROM AUTHOR]

We prove that the Cayley graphs X (G , S) and X + (G , S) are equienergetic for any abelian group G and any symmetric subset S. We then focus on the family of unitary Cayley graphs G R = X (R , R ⁎) , where R is a finite commutative ring with identity. We show that under mild conditions, { G R , G R + } are pairs of integral equienergetic non-isospectral graphs (generically connected and non-bipartite). Then, we obtain conditions such that { G R , G ¯ R } are equienergetic non-isospectral graphs. Finally, we characterize all integral equienergetic non-isospectral triples { G R , G R + , G ¯ R } such that all the graphs are also Ramanujan. [ABSTRACT FROM AUTHOR]

Several topologies can be defined on the prime, the maximal and the minimal prime spectra of a commutative ring; among them, we mention the Zariski topology, the patch topology and the flat topology. By using these topologies, Tarizadeh and Aghajani obtained recently new characterizations of various classes of rings: Gelfand rings, clean rings, absolutely flat rings, mp - rings, etc. The aim of this paper is to generalize some of their results to quantales, structures that constitute a good abstractization for lattices of ideals, filters and congruences. We shall study the flat and the patch topologies on the prime, the maximal and the minimal prime spectra of a coherent quantale. By using these two topologies one obtains new characterization theorems for hyperarchimedean quantales, normal quantales, B-normal quantales, mp - quantales and PF - quantales. The general results can be applied to several concrete algebras: commutative rings, bounded distributive lattices, MV-algebras, BL-algebras, residuated lattices, commutative unital l - groups, etc. [ABSTRACT FROM AUTHOR]

Recently it has been proved that if σ ∈ GL n (R) where R is a commutative ring and n ≥ 3 , then each of the matrices t k l (σ i j) (i ≠ j , k ≠ l) is a product of eight E n (R) -conjugates of σ and σ − 1. In this article we show that similar results hold true if R is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring. [ABSTRACT FROM AUTHOR]

Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph G called the (t , r) broadcast domination number which depends on the positive integer parameters t and r. In this setting, a v ∈ V is a broadcast vertex of transmission strength t if it transmits a signal of strength t − d (u , v) to every vertex u ∈ V with d (u , v) < t. Given a set of broadcast vertices S ⊆ V , the reception at vertex u is the sum of the transmissions from the broadcast vertices in S. The set S ⊆ V is called a (t , r) broadcast dominating set if every vertex u ∈ V has a reception strength r (u) ≥ r and for a finite graph G the cardinality of a smallest broadcast dominating set is called the (t , r) broadcast domination number of G. In this paper, we consider the infinite triangular grid graph and define efficient (t , r) broadcast dominating sets as those broadcasts that minimize signal waste. Our main result constructs efficient (t , r) broadcasts on the infinite triangular grid graph for all t ≥ r ≥ 1. Using these broadcasts, we then provide upper bounds for the (t , r) broadcast domination numbers for triangular matchstick graphs when (t , r) ∈ { (2 , 1) , (3 , 1) , (3 , 2) , (4 , 1) , (4 , 2) , (4 , 3) , (t , t) }. [ABSTRACT FROM AUTHOR]

Centrosymmetric matrices of order n over an arbitrary algebra R form a subalgebra of the full n × n matrix algebra over R. It is called the centrosymmetric matrix algebra of order n over R and denoted by S n (R). We prove (1) S n (R) is Morita equivalent to S 2 (R) if n is even, and to S 3 (R) if n ≥ 3 is odd; (2) the full n × n matrix algebra over R is a separable Frobenius extension of S n (R) ; and (3) if R is a commutative ring, then S n (R) is a cellular R -algebra in the sense of Graham-Lehrer for all n ≥ 1. [ABSTRACT FROM AUTHOR]

We study several aspects of the weak normalization in a graded extension of commutative rings. Applied to the Rees algebra of an ideal in a noetherian ring, the results obtained allow us to prove several properties of the weak subintegral closure of an ideal, a concept introduced by Vitulli and Leahy. The same methods are also used to eliminate one of the hypotheses in a theorem of Gaffney and Vitulli. [ABSTRACT FROM AUTHOR]

Over an arbitrary commutative ring, correspondences among three sets, the set of trace ideals, the set of stable ideals, and the set of birational extensions of the base ring, are studied. The correspondences are well-behaved, if the base ring is a Gorenstein ring of dimension one. It is shown that with one extremal exception, the surjectivity of one of the correspondences characterizes the Gorenstein property of the base ring, provided it is a Cohen-Macaulay local ring of dimension one. Over a commutative Noetherian ring, a characterization of modules in which every submodule is a trace module is given. The notion of anti-stable rings is introduced, exploring their basic properties. [ABSTRACT FROM AUTHOR]

Let k be a commutative ring and A a commutative k -algebra. In this paper we introduce the notion of enveloping algebra of Hasse–Schmidt derivations of A over k and we prove that, under suitable smoothness hypotheses, the canonical map from the above enveloping algebra to the ring of differential operators D A / k is an isomorphism. This result generalizes the characteristic 0 case in which the ring D A / k appears as the enveloping algebra of the Lie-Rinehart algebra of the usual k -derivations of A provided that A is smooth over k. [ABSTRACT FROM AUTHOR]

A commutative ring with identity is called a zip ring if each of its faithful ideals contains a finitely generated faithful ideal. A natural generalization of this notion is to require every faithful ideal to contain a countably generated faithful ideal. This is what we do in this paper. If a ring has this property, we call it a czip ring. Clearly, every zip ring is a czip ring. We give several characterizations of czip rings, including some in terms of subspaces of the space of minimal prime ideals with the Zariski topology. In rings of continuous functions we characterize the Tychonoff spaces X for which C (X) is a czip ring. This enables us to show that the class of czip rings properly contains that of zip rings. [ABSTRACT FROM AUTHOR]

In this study, we define the structure formation of the annihilator monic prime graph of commutative rings, whose distinct vertices X and J satisfies a condition a n n X J ≠ a n n X ⋃ a n n (J) , graph is denoted by AMPG (Z n [ x ] / 〈 f x 〉). The chromatic number of the annihilator monic prime graph is determined to establish the color (signless laplacian) energy of the graph. Also, we introduce color based energy of the annihilator monic prime graph, namely color distance signless Laplacian energy. Diag (T r) + D (G) + A χ , where Diag (T r) denotes the diagonal matrix of the vertex transmissions in the graph, D (G) is the distance matrix of vertices whose diagonals are zero, the entries of the matrix A χ are 1 if η i and η j are adjacent, −1 if η i and η j are non-adjacent with different colors, otherwise 0. The eigenvalues of color distance signless Laplacian matrix will be denoted by ∂ 1 χ + ≤ ∂ 2 χ + ≤ ∂ 3 χ + ≤ ⋯ ≤ ∂ n χ + , then DS L χ + of annihilator monic prime graph are define as E D S L χ + Z n x 〈 f x 〉 = ∑ 1 n ′ ∂ i χ + - t G where t (G) is the transmission of the graph. Based on the eigenvalues, we can study some more properties of graphs. Applications of color based energy are also discuss in this paper. [ABSTRACT FROM AUTHOR]

In this paper we define and study the mz -elements of an algebraic quantale as an abstraction of the mz -ideals of a commutative ring, recently introduced by Ighedo and McGovern. Using a result of Banaschewski, we prove that the set zA of the mz -elements of a coherent quantale A is a coherent frame, as the image of a localic nucleus s : A → A. We show that s : A → z A is a codense coherent quantale morphism, then we use the morphism s in order to obtain the quantale generalizations of some results obtained by Ighedo and McGovern. We study the relationship between the mz -elements of a coherent quantale A and the z -elements of the reticulation L (A) of A. In particular, we prove that the frame zA is isomorphic with the frame Z I d (L (A)) of the z -ideals of the lattice L (A). [ABSTRACT FROM AUTHOR]

For all subgroups H of a cyclic p -group G we define norm functors that build a G -Mackey functor from an H -Mackey functor. We give an explicit construction of these functors in terms of generators and relations based solely on the intrinsic, algebraic properties of Mackey functors and Tambara functors. We use these norm functors to define a monoidal structure on the category of Mackey functors where Tambara functors are the commutative ring objects. [ABSTRACT FROM AUTHOR]

A zero-nonzero pattern A is a matrix with entries from the set { 0 , ⁎ }. A square pattern A is spectrally arbitrary over R , a commutative ring with unity, if for each n -th degree monic polynomial f (x) ∈ R [ x ] , there exists a matrix A over R with pattern A such that the characteristic polynomial p A (x) = f (x). A pattern A is relaxed spectrally arbitrary over R if for each n -th degree monic polynomial f (x) ∈ R [ x ] , there exists a matrix A over R with either pattern A or a subpattern of A such that the characteristic polynomial p A (x) = f (x). We evaluate how the structure of rings affects how we determine if a pattern is spectrally arbitrary. We consider whether a pattern A that is spectrally arbitrary over R is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring S with unity. In particular, we establish that a pattern that is spectrally arbitrary over Z is relaxed spectrally arbitrary over Z m for all m ∈ Z + and spectrally arbitrary over Q. Similarly, a pattern that is spectrally arbitrary over the p -adic integers is spectrally arbitrary over the p -adic numbers. [ABSTRACT FROM AUTHOR]

We say a graph G is locally P if for each vertex v in G the open neighbourhood of v induces a graph with property P. The Hamilton Cycle Problem (HCP) is the problem of deciding whether a graph contains a Hamilton cycle. It is known that the HCP is NP-complete for locally traceable (LT) graphs with maximum degree 6. We extend that result to 1-tough graphs and to graphs with restricted degree sequences. If R is a set of nonnegative integers, we say a graph G is R -regular if the degrees of all the vertices in V (G) are elements of R. We show that the HCP is NP-complete for R -regular LT graphs if R is any set of natural numbers with m a x (R) ≥ 6 , with the possible exception of { 4 , 6 } and { 6 }. It is known that the HCP is NP-complete for LH graphs with maximum degree 10. We improve this result by showing that the HCP is NP-complete for 1-tough LH graphs with maximum degree 9 and for R -regular LH graphs if R is any set of natural numbers with min (R) = 3 and max (R) ∈ { 9 , 10 } , or if R is any set of natural numbers with max (R) ≥ 11. Finally, we show that the HCP for k -connected LH graphs that are also locally (k − 1) -connected is NP-complete for every k ≥ 3. [ABSTRACT FROM AUTHOR]

Let n = p 1 n 1 p 2 n 2 ⋯ p r n r , where r , n 1 , n 2 , ... , n r are positive integers and p 1 , p 2 , ... , p r are distinct prime numbers with p 1 < p 2 < ... < p r. For the finite cyclic group C n of order n , let P (C n) be the power graph of C n and κ (P (C n)) be the vertex connectivity of P (C n). It is known that κ (P (C n)) = p 1 n 1 − 1 if r = 1. For r ≥ 2 , we determine the exact value of κ (P (C n)) when 2 ϕ (p 1 p 2 ⋯ p r − 1) ≥ p 1 p 2 ⋯ p r − 1 , and give an upper bound for κ (P (C n)) when 2 ϕ (p 1 p 2 ⋯ p r − 1) < p 1 p 2 ⋯ p r − 1 , which is sharp for many values of n but equality need not hold always. [ABSTRACT FROM AUTHOR]

Abstract Let R be a commutative ring. We show that any complete duality pair gives rise to a theory of relative homological algebra, analogous to Gorenstein homological algebra. Indeed Gorenstein homological algebra over a commutative Noetherian ring of finite Krull dimension can be recovered from the duality pair (F , I) where F is the class of flat R -modules and I is the class of injective R -modules. For a general R , the AC-Gorenstein homological algebra of Bravo–Gillespie–Hovey is the one coming from the duality pair (L , A) where L is the class of level R -modules and A is class of absolutely clean R -modules. Indeed we show here that the work in [1] can be extended to obtain similar abelian model structures on R -Mod from any a complete duality pair (L , A). It applies in particular to the original duality pairs constructed by Holm–Jørgensen. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of classifying commutative rings according to the crosscap number of it's associating graphs. In particular, we enumerate all isomorphism classes of commutative rings with identity whose zero-divisor graph and total graph have crosscap two respectively. This answers a case of a conjecture posed by Chiang-Hsieh in (2008) and Khashyarmanesh et al. in (2013). [ABSTRACT FROM AUTHOR]

Ordered partitions of elements of a reduced abelian monoid are defined and studied by means of the solutions of linear diophantine equations. Links to feedback classification of linear dynamical systems over certain commutative rings are given in the same way as partitions of integers are related to feedback classification of linear dynamical systems over fields in the classical literature. [ABSTRACT FROM AUTHOR]

In this paper, results of Tsarelunga resp. Comfort, Szambien and the first-listed author are improved. Throughout this abstract, (R , T) denotes a nonmetrizable compact ring. First a main tool is shown: If (R , T) is topologically nilpotent, then w (R) = w (R / R 2 ‾) holds. By using tensor products of unitary modules it is proved that every nonmetrizable compact ring with an identity has a proper pseudocompact refinement. (R , T) admits exactly 2 2 | R | -many pseudocompact ring topologies on R finer than T in the following cases: R is a commutative local ring; (R , T) is topologically nilpotent; (R , T) is commutative such that w (R , T) is a regular cardinal number. [ABSTRACT FROM AUTHOR]

Abstract We compute the k th power-sums (for all k > 0) over an arbitrary finite unital ring R. This unifies and extends the work of Brawley et al. (1974) [1] for matrix rings, with folklore results for finite fields and finite cyclic groups, and more general recent results of Grau and Oller-Marcén (2017) [12] for commutative rings. As an application, we resolve a conjecture by Fortuny Ayuso et al. (2017) [7] on zeta values for matrix rings over finite commutative rings. We further recast our main result via zeta values over polynomial rings, and end by classifying the translation-invariant polynomials over a large class of finite commutative rings. [ABSTRACT FROM AUTHOR]

Abstract A ring R is said to be of stable range 1.5 if for each a , b ∈ R and 0 ≠ c ∈ R satisfying a R + b R + c R = R there exists r ∈ R such that (a + b r) R + c R = R. Let R be a commutative domain in which all finitely generated ideals are principal, and let R be of stable range 1.5. Then each matrix A over R is reduced to Smith's canonical form by transformations PAQ in which P and Q are invertible and at least one of them can be chosen to be a product of elementary matrices. We generalize Helmer's theorem about the greatest common divisor of entries of A over R. [ABSTRACT FROM AUTHOR]

Abstract Given a commutative ring A and a finitely generated ideal I , we prove that I ∞ -torsion A -modules that are also I -adically complete (or merely derived I -complete) must have bounded I ∞ -torsion, i.e., they are killed by I n for some n ≥ 0. [ABSTRACT FROM AUTHOR]

Abstract We prove that the Brauer group of the moduli stack of elliptic curves M 1 , 1 , k over an algebraically closed field k of characteristic 2 is isomorphic to Z / (2). We also compute the Brauer group of M 1 , 1 , k where k is a finite field of characteristic 2. [ABSTRACT FROM AUTHOR]

Abstract Let R be a finite commutative ring with unit element 1 ≠ 0 , and let G R denote the unitary one-matching bi-Cayley graph over R. In this paper, we determine the energies of G R and its line graph, respectively. We also characterize when G R as well as its line graph is hyperenergetic or hypoenergetic, respectively. [ABSTRACT FROM AUTHOR]

Abstract Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension pd (R / I) of R / I is at most 36, although the example with largest projective dimension he constructed has pd (R / I) = 5. Based on computational evidence, it had been conjectured that pd (R / I) ≤ 5. In the present paper we prove this conjectured sharp bound. [ABSTRACT FROM AUTHOR]

Abstract We show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s -divisible modules, where s is a regular element of a commutative ring R , is closed under flat covers then the quotient ring R / s R satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings. [ABSTRACT FROM AUTHOR]

Abstract Let R be a commutative Noetherian ring of dimension two with 1 / 2 ∈ R and let A = R [ X 1 , ⋯ , X n ]. Let P be a projective A -module of rank 2. In this article, we prove that P is cancellative if ∧ 2 (P) ⊕ A is cancellative. [ABSTRACT FROM AUTHOR]

Abstract We introduce the notion of syzygy for a set of reduction operators and relate it to the notion of syzygy for presentations of algebras. We give a method for constructing a linear basis of the space of syzygies for a set of reduction operators. We interpret these syzygies in terms of the confluence property from rewriting theory. This enables us to optimise the completion procedure for reduction operators based on a criterion for detecting useless reductions. We illustrate this criterion with an example of construction of commutative Gröbner basis. [ABSTRACT FROM AUTHOR]

Abstract A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R -modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R -modules. [ABSTRACT FROM AUTHOR]

We provide a class of commutative Noetherian domains R of dimension d such that every finitely generated projective R -module P of rank d splits off a free summand of rank one. On this class, we also show that P is cancellative. At the end we give some applications to the number of generators of a module over the Rees algebras. [ABSTRACT FROM AUTHOR]

The present paper is a sequel to our previous work on almost uniserial rings and modules, which appeared in the Journal of Algebra in 2016; it studies rings over which every (left and right) module is almost serial. A module is almost uniserial if any two of its submodules are either comparable in inclusion or isomorphic. And a module is almost serial if it is a direct sum of almost uniserial modules. The results of the paper are inspired by a characterization of Artinian serial rings as rings having all left (or right) modules serial. We prove that if R is a local ring and all left R -modules are almost serial then R is an Artinian ring which is uniserial either on the left or on the right. We also produce a connection between local rings having all left and right modules almost serial, local balanced rings studied by Dlab and Ringel and local Köthe rings. Finally we prove Morita invariance of the almost serial property and list some consequences. [ABSTRACT FROM AUTHOR]