Rings satisfying polynomial constraints

A semigroup S is called a left reductive semigroup if, for all elements a,b∈S, the assumption “xa=xb for all x∈S” implies a=b. A congruence α on a semigroup S is called a left reductive congruence if the factor semigroup S/α is left reductive. In this paper we deal with the left reductive congruences on semigroups. Let S be a semigroup and ϱ a congruence on S. Consider the sequence ϱ (0)⊆ϱ (1)⊆⋯⊆ϱ (n)⊆⋯ of congruences on S, where ϱ (0)=ϱ and, for an arbitrary non-negative integer n, ϱ (n+1) is defined by (a,b)∈ϱ (n+1) if and only if (xa,xb)∈ϱ (n) for all x∈S. We show that \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\subseteq \mathit{lrc}(\varrho )\) for an arbitrary congruence ϱ on a semigroup S, where lrc(ϱ) denotes the least left reductive congruence on S containing ϱ. We focuse our attention on congruences ϱ on semigroups S for which the congruence \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is left reductive. We prove that, for a congruence ϱ on a semigroup S, \(\bigcup_{i=0}^{\infty}\varrho^{(i)}\) is a left reductive congruence of S if and only if \(\bigcup_{i=0}^{\infty}\iota_{(S/\varrho)}^{(i)}\) is a left reductive congruence on the factor semigroup S/ϱ (here ι (S/ϱ) denotes the identity relation on S/ϱ). After proving some other results, we show that if S is a Noetherian semigroup (which means that the lattice of all congruences on S satisfies the ascending chain condition) or a semigroup in which S n =S n+1 is satisfied for some positive integer n then the universal relation on S is the only left reductive congruence on S if and only if S is an ideal extension of a left zero semigroup by a nilpotent semigroup. In particular, S is a commutative Noetherian semigroup in which the universal relation on S is the only left reductive congruence on S if and only if S is a finite commutative nilpotent semigroup.

For any commutative semigroup S and positive integer m the power function f: S rightarrow S defined by f(x) = x^m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

We investigate the intersection problem for finite semigroups, which asks for a given set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. We introduce compressibility measures as a useful tool to classify the intersection problem for certain classes of finite semigroups into circuit complexity classes and Turing machine complexity classes. Using this framework, we obtain a new and simple proof that for groups and commutative semigroups, the problem is contained in \(\mathsf {NP}\). We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove \(\mathsf {NP}\)-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply containment in the circuit complexity class \(\mathsf {qAC}^k\) (for some \(k \in \mathbb {N}\)) and decidability in quasi-polynomial time. We also establish connections to the monoid variant of the problem.

A semigroup S is called left (right) weakly commutative if, for every a, b ∈ S, there exist x ∈ S and a positive integer n such that (ab) n = bx ((ab) n = xa). A semigroup which is both left and right weakly commutative is called a weakly commutative semigroup. In this chapter we deal with left weakly commutative, right weakly commutative and weakly commutative semigroups. It is shown that a semigroup is a semilattice of left archimedean (right archimedean, t-archimedean) semigroups if and only if it is right weakly commutative (left weakly commutative, weakly commutative). It is proved that a weakly commutative 0-simple semigroup is a group with a zero adjoined. Moreover, a semigroup is weakly commutative archimedean and contains an idempotent element if and only if it is an ideal extension of a group by a nil semigroup. We get, as a consequence, that a semigroup is weakly commutative and regular if and only if it is a Clifford semigroup. We show that a right (left) weakly commutative semigroup is embeddable into a group if and only if it is cancellative. At the end of the chapter we deal with the least weakly separative congruence on weakly commutative semigroups. It is proved that if 5 is a left weakly commutative semigroup then σ defined by a σ b if and only if ab n = bn+1 and ba n = an+1 for a positive integer n is a weakly separative congruence on 5. Similarly, if S is a right weakly commutative semigroup then r defined by a τ b if and only if b n a = b n+1 and a n b = an+1 for some positive integer n is a weakly separative congruence on S. Moreover, π = σ ∩ τ is the least weakly separative congruence on a weakly commutative semigroup.

From Theorem 19.1 of [l] it follows that the lattice of varieties (equational classes) of Abelian groups is distributive.We show that the lattice of varieties of commutative semigroups is not modular by considering an example from the dual lattice of closed commutative semigroup laws.We will assume and use without comment that any given set of laws contains the commutative and associative laws.The binary operation is denoted by juxtaposition.By (a, b), where a and b are positive integers, we will mean the term xayb.Let the sets A', B', and C of laws be given by the following table.

An associative ring R is called a unique addition ring (UA-ring) if its multiplicative semigroup (R, · ) can be equipped with a unique binary operation+ transforming the triple (R, ·, +) to a ring. An R-module A is said to be an End-UA-module if the endomorphism ring EndR(A) of A is a UA-ring. In the paper, the torsion-free End-UA-modules over commutative Dedekind domains are studied. In some classes of Abelian torsion-free groups, the Abelian groups having UA-endomorphism rings are found.

A semigroup S is called a weakly commutative semigroup if, for every a,b∈S, there is a positive integer n such that (ab) n ∈Sa∩bS. A semigroup S is called archimedean if, for every a,b∈S, there are positive integers m and n such that a n ∈SbS and b m ∈SaS. It is known that every weakly commutative semigroup is a semilattice of weakly commutative archimedean semigroups. A semigroup S is called a weakly separative semigroup if, for every a,b∈S, the assumption a 2=ab=b 2 implies a=b. In this paper we show that a weakly commutative semigroup is weakly separative if and only if its archimedean components are weakly cancellative. This result is a generalization of Theorem 4.16 of Clifford and Preston (The Algebraic Theory of Semigroups, Am. Math. Soc., Providence, 1961).

Abstract. A right ideal I of a ring R is small in case for every proper right ideal Kof R, K + I 6= R. A right R-module M is called PS-injective if every R-homomorphismf : aR ! M for every principally small right ideal aR can be extended to R ! M. A ringR is called right PS-injective if R is PS-injective as a right R-module. We develop, inthis article, PS-injectivity as a generalization of P-injectivity and small injectivity. Manycharacterizations of right PS-injective rings are studied. In light of these facts, we getseveral new properties of a right GPF ring and a semiprimitive ring in terms of rightPS-injectivity. Related examples are given as well. 1. IntroductionThroughout this paper, R is an associative ring with identity and all modulesare unitary. Let R be a ring. The Jacobson radical and nil radical of R are denotedby J(R) and Nil(R), respectively. The right singular ideal is denoted by Z(R R ),the socles are denoted by soc(R R ) and soc( R R). If X is a subset of R, the right(resp. left) annihilator of X in R is denoted by r

Associative rings with metabelian adjoint group

The aim of this paper is to classify all monogenic ternary semigroups, up to isomorphism. We divide them to two groups: finite and infinite. We show that every infinite monogenic ternary semigroup is isomorphic to the ternary semigroup O, the odd positive integers with ordinary addition. Then we prove that all finite monogenic ternary semigroups with the same index and the same period are isomorphic. We also investigate structure of finite monogenic ternary semigroups and we prove that any finite monogenic ternary semigroup is isomorphic to a quotient ternary semigroup.

In this paper, we introduce the concepts of derivation of degree n, generalized derivation of degree n and ternary derivation of degree n, where n is a positive integer, and then we study the algebraic properties of these mappings. For instance, we study the image of derivations of degree n on algebras and in this regard we prove that, under certain conditions, every derivation of degree n on an algebra maps the algebra into its Jacobson radical. Also, we present some characterizations of these mappings on algebras. For example, under certain assumptions, we show that if f is an additive generalized derivation of degree n with an associated mapping d, then either f is a linear generalized derivation with the associated linear derivation d or f and d are identically zero. Some other related results are also established.

An element a a of a ring R R is a uniquely clean element if a a can be uniquely expressed as a sum of an idempotent and a unit in R R . For a positive integer n n , a ring R R is called uniquely n n -clean if a n a^n is uniquely clean for all a ∈ R a\in R . The ring R R is called uniquely π \pi -clean if for each a ∈ R a\in R there is a positive integer n ( a ) n(a) such that a n ( a ) a^{n(a)} is uniquely clean. It is proved that a ring R R with Jacobson radical J ( R ) J(R) is uniquely π \pi -clean iff R / J ( R ) R/J(R) is a periodic ring, idempotents of R R are central and idempotents lift modulo J ( R ) J(R) . An analogous characterization of uniquely n n -clean rings in terms of J J -rings is obtained. Consequences of these results are discussed and, in particular, the known results on uniquely clean rings are reproved by simply letting n = 1 n=1 .

On group automata over finite fields

The intersection problem for finite semigroups asks, given a set of regular languages, represented by recognizing morphisms to finite semigroups, whether there exists a word contained in their intersection. In previous work, it was shown that is problem is [Formula: see text]-complete. We introduce compressibility measures as a useful tool to classify the complexity of the intersection problem for certain classes of finite semigroups. Using this framework, we obtain a new and simple proof that for groups and for commutative semigroups, the problem (as well as the variant where the languages are represented by finite automata) is contained in [Formula: see text]. We uncover certain structural and non-structural properties determining the complexity of the intersection problem for varieties of semigroups containing only trivial submonoids. More specifically, we prove [Formula: see text]-hardness for classes of semigroups having a property called unbounded order and for the class of all nilpotent semigroups of bounded order. On the contrary, we show that bounded order and commutativity imply decidability in poly-logarithmic time on alternating random-access Turing machines with a single alternation. We also establish connections to the monoid variant of the problem.

Continuing an earlier examination of the codominant dimension of rings and modules, a categorical characterization is given for rings of equivalent dominant and codominant dimensions. Specifically, the question is reduced to when R R and the left minimal injective cogenerator R can be used as test modules respectively for the dominant dimension of the projective modules and the codominant dimension of the injective modules. These conditions are in turn characterized by when the injective projective modules are Z-injective. Also, a new and shortened version is given for the proof that the codominant dimension of the injective modules is equal to the dominant dimension of the projective modules. Throughout, R is an associative ring with unity and all modules considered are unital. We denote a minimal injective cogenerator for R)R by R such being isomorphic to an injective hull of a direct sum of a complete representative class of simple left R-modules. For a left perfect ring R and a left R-module RM we define the codominant dimension of RM, denoted codom dim RM, to be greater than or equal to n if for a minimal projective resolution . P* P M 0, Pi is injective for i = 1, 29, . . ., n. The left codominant dimesnion of R is defined to be codom dim RU. The dominant dimension of RM is defined in a dual manner, and the left dominant dimension of R is dom dimRR. Further, for R left perfect, we define codom dimR9 to be infIcodom dim RE: RE injectivel. Similarly, dom dim Ris given by inf dom dimRP: RP projectivel. Finally, we say that RE (nRP) is a test module for RI (RT) provided codom dim RE= codom dimRI (dom dimR P = dom dimRY). In [2] we showed that dom dim R 9= codom dim RI. As an application it can Presented to the Society, January 25, 1973; received by the editors April 16, 1973. AMS (MOS) subject classifications (1970). Primary 16A60; Secondary 16A50, 16A52.