Structure of free strong doppelsemigroups

We consider strong doppelsemigroups which are sets with two binary associative operations satisfying axioms of strong interassociativity. Commutative dimonoids in the sense of Loday are examples of strong doppelsemigroups and two strongly interassociative semigroups give rise to a strong doppelsemigroup. The main aim of this paper is to construct a free strong doppelsemigroup, a free n-dinilpotent strong doppelsemigroup, a free commutative strong doppelsemigroup and a free n-nilpotent strong doppelsemigroup. We also characterize the least n-dinilpotent congruence, the least commutative congruence, the least n-nilpotent congruence on a free strong doppelsemigroup and establish that the automorphism group of every constructed free algebra is isomorphic to the symmetric group.     

Free left n-dinilpotent doppelsemigroups

A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain equations. Commutative dimonoids in the sense of Loday are examples of doppelsemigroups and two interassociative semigroups give rise to a doppelsemigroup. We introduce left (right) n-dinilpotent doppelsemigroups which are analogs of left (right) nilpotent semigroups of rank n considered by Schein. A free left (right) n-dinilpotent doppelsemigroup is constructed and the least left (right) n-dinilpotent congruence on a free doppelsemigroup is characterized. We also establish that the semigroups of the free left (right) n-dinilpotent doppelsemigroup are isomorphic and the automorphism group of the free left (right) n-dinilpotent doppelsemigroup is isomorphic to the symmetric group.     

On lattices embeddable into subsemigroup lattices. III: Nilpotent semigroups

We prove that the class of the lattices embeddable into subsemigroup lattices of n-nilpotent semigroups is a finitely based variety for all n < ω. Repnitski? showed that each lattice embeds into the subsemigroup lattice of a commutative nilsemigroup of index 2. In this proof he used a result of Bredikhin and Schein which states that each lattice embeds into the suborder lattices of an appropriate order. We give a direct proof of the Repnitski? result not appealing to the Bredikhin-Schein theorem, so answering a question in a book by Shevrin and Ovsyannikov.     

Variations on average character degrees and p-nilpotence

We prove that if p is an odd prime, G is a solvable group, and the average value of the irreducible characters of G whose degrees are not divisible by p is strictly less than 2(p + 1)/(p + 3), then G is p-nilpotent. We show that there are examples that are not p-nilpotent where this bound is met for every prime p. We then prove a number of variations of this result.     

Subdirectly irreducible commutative multiplicatively idempotent semirings

Commutative multiplicatively idempotent semirings were studied by the authors and F. ?vr?ek, where the connections to distributive lattices and unitary Boolean rings were established. The variety of these semirings has nice algebraic properties and hence there arose the question to describe this variety, possibly by its subdirectly irreducible members. For the subvariety of so-called Boolean semirings, the subdirectly irreducible members were described by F. Guzmán. He showed that there were just two subdirectly irreducible members, which are the 2-element distributive lattice and the 2-element Boolean ring. We are going to show that although commutative multiplicatively idempotent semirings are at first glance a slight modification of Boolean semirings, for each cardinal n > 1, there exist at least two subdirectly irreducible members of cardinality n and at least 2ⁿ such members if n is infinite. For \({n \in \{2, 3, 4\}}\) the number of subdirectly irreducible members of cardinality n is exactly 2.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

Classification of rings with toroidal Jacobson graph

Let R be a commutative ring with nonzero identity and J(R) the Jacobson radical of R. The Jacobson graph of R, denoted by J_R, is defined as the graph with vertex set RJ(R) such that two distinct vertices x and y are adjacent if and only if 1 ? xy is not a unit of R. The genus of a simple graph G is the smallest nonnegative integer n such that G can be embedded into an orientable surface S_n. In this paper, we investigate the genus number of the compact Riemann surface in which J_R can be embedded and explicitly determine all finite commutative rings R (up to isomorphism) such that J_R is toroidal.     

Some results on the annihilator graph of a commutative ring

Let R be a commutative ring. The annihilator graph of R, denoted by AG(R), is the undirected graph with all nonzero zero-divisors of R as vertex set, and two distinct vertices x and y are adjacent if and only if ann _R (xy) ≠ ann _R (x) ∪ ann _R (y), where for z ∈ R, ann _R (z) = {r ∈ R: rz = 0}. In this paper, we characterize all finite commutative rings R with planar or outerplanar or ring-graph annihilator graphs. We characterize all finite commutative rings R whose annihilator graphs have clique number 1, 2 or 3. Also, we investigate some properties of the annihilator graph under the extension of R to polynomial rings and rings of fractions. For instance, we show that the graphs AG(R) and AG(T(R)) are isomorphic, where T(R) is the total quotient ring of R. Moreover, we investigate some properties of the annihilator graph of the ring of integers modulo n, where n ? 1.     

Free <Emphasis Type="Italic">n</Emphasis>-Tuple Semigroups

The present paper is devoted to the study of n-tuple semigroups. A free n-tuple semigroup of arbitrary rank is constructed and, as a consequence, singly generated free n-tuple semigroups are characterized. Moreover, examples of n-tuple semigroups are presented, the independence of the n-tuple semigroup axioms is proved, and it is shown that the natural semigroups of the constructed free n-tuple semigroup are isomorphic and the automorphism group of this n-tuple semigroup is isomorphic to a symmetric group.     

Equalizers and kernels in categories of monoids

In dealing with monoids, the natural notion of kernel of a monoid morphism \(f:M\rightarrow N\) between two monoids M and N is that of the congruence \(\sim _f\) on M defined, for every \(m,m'\in M\), by \(m\sim _fm'\) if \(f(m)=f(m')\). In this paper, we study kernels and equalizers of monoid morphisms in the categorical sense. We consider the case of the categories of all monoids, commutative monoids, cancellative commutative monoids, reduced Krull monoids, inverse monoids and free monoids. In all these categories, the kernel of \(f:M\rightarrow N\) is simply the embedding of the submonoid \(f^{-1}(1_N)\) into M, but a complete characterization of kernels in these categories is not always trivial, and leads to interesting related notions.     

On (<Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-absorbing ideals of commutative rings

Let R be a commutative ring with 1 ≠ 0 and U(R) be the set of all unit elements of R. Let m, n be positive integers such that m > n. In this article, we study a generalization of n-absorbing ideals. A proper ideal I of R is called an (m, n)-absorbing ideal if whenever a ₁?a _m ∈I for a ₁,…, a _m ∈R?U(R), then there are n of the a _i ’s whose product is in I. We investigate the stability of (m, n)-absorbing ideals with respect to various ring theoretic constructions and study (m, n)-absorbing ideals in several commutative rings. For example, in a Bézout ring or a Boolean ring, an ideal is an (m, n)-absorbing ideal if and only if it is an n-absorbing ideal, and in an almost Dedekind domain every (m, n)-absorbing ideal is a product of at most m ? 1 maximal ideals.     

Subsemigroup,ideal and congruence growth of free semigroups

The growth of cofinite subsemigroups of free semigroups is investigated. Lower and upper bounds for the sequence are given and it is shown to have superexponential growth of strict type nⁿ for finite free rank greater than 1. Ideal growth is shown to be exponential with strict type 2ⁿ and congruence growth is shown to be at least exponential. In addition we consider the case when the index is fixed and rank increasing, proving that for subsemigroups and ideals this sequence fits a polynomial of degree the index, whereas for congruences this fits an exponential equation of base the index. We use these results to describe an algorithm for computing values of these sequences and give a table of results for low rank and index.     

Locally Unmixed Modules and Linearly Equivalent Ideal Topologies

Let R be a commutative Noetherian ring, and let N be a non-zero finitely generated R-module. The purpose of this paper is to show that N is locally unmixed if and only if, for any N-proper ideal I of R generated by ht _N I elements, the topology defined by (I N)⁽ⁿ⁾, n ≥ 0, is linearly equivalent to the I-adic topology.     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ _α ⁰ -categorical integral domain (commutative semigroup) which is not relatively Δ _α ⁰ -categorical (i.e., no formally Σ _α ⁰ Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ _α ⁰ -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ _α ⁰ -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ _α ⁰ -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ _α ⁰ (X) is not Δ _α ⁰ . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.     

Local derivations of full matrix rings

Let \({\mathcal{R}}\) be a unital commutative ring and \({\mathcal{M}}\) be a 2-torsion free central \({\mathcal{R}}\) -bimodule. In this paper, for \({n \geqq 3}\), we show that every local derivation from M _n (\({\mathcal{R}}\)) into M _n (\({\mathcal{M}}\)) is a derivation.     

On a functional equation related to derivations in prime rings

In this paper we prove the following result. Let m ≥ 1, n ≥ 1 be fixed integers and let R be a prime ring with m + n + 1 ≤ char(R) or char(R) = 0. Suppose there exists an additive nonzero mapping D : R → R satisfying the relation 2D(x ^n+m+1) = (m + n + 1)(x ^m D(x)x ⁿ + x ⁿ D(x)x ^m ) for all \({x\in R}\). In this case R is commutative and D is a derivation.     

Multiple commutator formulas for unitary groups

Let (A,Λ) be a formring such that A is quasi-finite R-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak’s unitary groups GU(2n, A, Λ), n ≥ 3. For a form ideal (I, Γ) of the form ring (A, Λ) we denote by EU(2n, I, Γ) and GU(2n, I, Γ) the relative elementary group and the principal congruence subgroup of level (I, Γ), respectively. Now, let (I _i , Γ _i ), i = 0,...,m, be form ideals of the form ring (A, Λ). The main result of the present paper is the following multiple commutator formula: [EU(2n, I ₀, Γ ₀),GU(2n, I ₁, Γ ₁), GU(2n, I ₂, Γ ₂),..., GU(2n, I _m , Γ _m )] =[EU(2n, I ₀, Γ ₀), EU(2n, I ₁, Γ ₁), EU(2n, I ₂, Γ ₂),..., EU(2n, I _m , Γ _m )], which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classicallike groups over commutative and finite-dimensional rings.     

Compositions,derivations and polynomials

Let R be a prime ring with extended centroid C. In this paper, we discuss the case when the composition of a generalized derivation δ and a polynomial map f(Y) ∈ C[Y] of R is commutative on a non-zero right ideal ρ and a non-commutative Lie ideal L of R respectively, i.e., when the identity δ ○ f(x) = f ○ δ(x) holds on ρ or L. As applications of our main theorems, we clarify the generalized derivations which act as n-Jordan homomorphisms (S _n -homomorphisms) on ρ or L.     

Sum-free sets in abelian groups

An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS _i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? _i=1 ^k S _i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z ^t →Z ^d is a polynomial vanishing at zero,T is a measure preservingZ ^d action andn is an IP system intoZ ^t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ ^d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).