ARMENDARIZ RINGS RELATIVE TO A MONOID

ABSTRACT

For a monoid M, we introduce M-Armendariz rings, which are generalizations of Armendariz rings; and we investigate their properties. Every reduced ring is M-Armendariz for any unique product monoid M. We show that if R is a reduced and M-Armendariz ring, then R is M × N-Armendariz, where N is a unique product monoid. It is also shown that a finitely generated Abelian group G is torsion free if and only if there exists a ring R such that R is G-Armendariz. Moreover, we study the relationship between the Baerness and the PP-property of a ring R and those of the monoid ring R[M] in case R is M-Armendariz.     

Annihilator Properties of Skew Monoid Rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u ₁,…, u _t } with 0 added, and M a factor of F setting certain monomial in U to 0, enough so that, for some n, M ⁿ  = 0. In this article, we study various annihilator properties and a variety of conditions and related properties that the skew monoid ring R[M; σ] is inherited from R.     

Baer and Quasi-Baer Properties of Skew Generalized Power Series Rings

Let R be a ring, (S, ≤) a strictly ordered monoid and ω: S → End(R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal'cev–Neumann Laurent series rings. In this article, we study relations between the (quasi-) Baer, principally quasi-Baer and principally projective properties of a ring R, and its skew generalized power series extension R[[S, ω]]. As particular cases of our general results, we obtain new theorems on (skew) group rings, Mal'cev–Neumann Laurent series rings, and the ring of generalized power series.     

Zero-Divisors,Torsion Elements,and Unions of Annihilators

Let R be a commutative ring and M be a nonzero R-module. Now Z(M) = {r ∈ R | rm = 0 for some 0 ≠ m ∈ M} is a union of prime ideals of R and T(M) = {m ∈ M | rm = 0 for some 0 ≠ r ∈ R} is a union of prime submodules of M if M ≠ T(M). We investigate representations of Z(M) and T(M) as unions of primes each of which is a union of annihilators.     

Radicals of Skew Polynomial and Skew Laurent Polynomial Rings Over Skew Armendariz Rings

In this note we study radicals of skew polynomial ring R[x; α] and skew Laurent polynomial ring R[x, x ^?1; α], for a skew-Armendariz ring R. In particular, among the other results, we show that for an skew-Armendariz ring R, J(R[x; α]) = N ₀(R[x; α]) = Ni?_*(R)[x; α] and J(R[x, x ^?1; α]) = N ₀(R[x, x ^?1; α]) = Ni?_*(R)[x, x ^?1; α].     

Principally Quasi-Baer skew Generalized Power Series modules

Let R be a ring and S a strictly totally ordered monoid. Let ω: S → End(R) be a monoid homomorphism. Let M _R be an ω-compatible module and either R satisfies the ascending chain conditions (ACC) on left annihilator ideals or every S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join. We show that M _R is p.q.-Baer if and only if the generalized power series module M[[S]] _{R[[S, ω]]} is p.q.-Baer. As a consequence, we deduce that for an ω-compatible ring R, the skew generalized power series ring R[[S, ω]] is right p.q.-Baer if and only if R is right p.q.-Baer and either R satisfies the ACC on left annihilator ideals or any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. Examples to illustrate and delimit the theory are provided.     

On Skew Inverse Laurent-Serieswise Armendariz Rings

We study the skew inverse Laurent-serieswise Armendariz (or simply, SIL-Armendariz) condition on R, a generalization of the standard Armendariz condition from polynomials to skew inverse Laurent series. We study relations between the set of annihilators in R and the set of annihilators in R((x ^?1; α)). Among applications, we show that a number of interesting properties of a SIL-Armendariz ring R such as the Baer and the α-quasi Baer property transfer to its skew inverse Laurent series extensions R((x ^?1; α)) and vice versa. For an α-weakly rigid ring R, R((x ^?1; α)) is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S _?(R) has a generalized countable join in R. Various types of examples of SIL-Armendariz rings is provided.     

Minimalities and Modules over Dedekind-Like Rings

We are interested in (right) modules M satisfying the following weak divisibility condition: If R is the underlying ring, then for every r ∈ R either Mr = 0 or Mr = M. Over a commutative ring, this is equivalent to say that M is connected with regular generics. Over arbitrary rings, modules which are “minimal” in several model theoretic senses satisfy this condition. In this article, we investigate modules with this weak divisibility property over Dedekind-like rings and over other related classes of rings.     

Cohen–Macaulay Modules in Dimension >s and Results on Local Cohomology

Let (R,𝔪) be a local ring and s ≥ ?1. Using the notion of M-sequence in dimension > s, we introduce Cohen–Macaulay modules in dimension > s. Among other things concerning Cohen–Macaulay modules in dimension > s, some finiteness results of the support and the associated primes of local cohomology modules are investigated.     

Quasi-Armendariz Rings Relative to a Monoid

A ring R is called an M-quasi-Armendariz ring (a quasi-Armendariz ring relative to a monoid M) if whenever elements α = a ₁ g ₁ + a ₂ g ₂ + ··· + a _n g _n , β = b ₁ h ₁ + b ₂ h ₂ + ··· + b _m h _m  ? R[M] satisfy α R[M]β = 0, then a _i Rb _j  = 0 for each i, j. After discussing some basic properties of M-quasi-Armendariz rings, we consider the influence of transformation of the monoid M and the ring R on this property. Particularly, we give some sufficient conditions for the monoids M, N, and the ring R under which R is M × N-quasi-Armendariz if and only if R is M-quasi-Armendariz and N-quasi-Armendariz.     

On Induced Modules Over Ore Extensions

Abstract

Let R be a ring and S = R[x;σ, δ] its Ore extension. For an R-module M _R we investigate the uniform dimension and associated primes of the induced S-module M ? _R  S.     

Examples of Generic Lattice Ideals of Codimension 3

A commutative ring R is said to be strongly Hopfian if the chain of annihilators ann(a) ? ann(a ²) ? … stabilizes for each a ∈ R. In this article, we are interested in the class of strongly Hopfian rings and the transfer of this property from a commutative ring R to the ring of the power series R[[X]]. We provide an example of a strongly Hopfian ring R such that R[[X]] is not strongly Hopfian. We give some necessary and sufficient conditions for R[[X]] to be strongly Hopfian.     

Annihilator Primes and Foundation Primes

ABSTRACT

Let R be a Noetherian ring and M a finitely generated R -module. In this article, we introduce the set of prime ideals Fnd  M , the foundation primes of M . Using the fact that this set is nicely organized by foundation levels, we present an approach to the problem of understanding Annspec  M , the annihilator primes of M , via Fnd  M . We show: (1) Fnd  M is a finite set containing Annspec  M . Further, suppose that moreover every ideal of R has a centralizing sequence of generators; now, Annspec  M is equal to the set Ass  M of associated primes of M. Then: (2) For an arbitrary P  ∈ Fnd  M , P  ∈ Annspec  M if and only if there is no Q  ∈ Annspec  M such that P contains Q , and at the same time, the minimal foundation level on which appears P is greater than the minimal foundation level on which appears Q .     

Radicals of Ore Extension of Skew Armendariz Rings

Let R be a ring with an endomorphism α and an α-derivation δ. In this article, for a skew-Armendariz ring R we study some properties of skew polynomial ring R[x; α, δ]. In particular, among other results, we show that for an (α, δ)-compatible skew-Armendariz ring R, γ(R[x; α, δ]) = γ(R)[x; α, δ] = Ni?_*(R)[x; α, δ], where γ is a radical in the class of radicals which includes the Wedderburn, lower nil, Levitzky, and upper nil radicals. We also show that several properties, including the symmetric, reversible, ZC_n, zip, and 2-primal property, transfer between R and the skew polynomial ring R[x; α, δ], in case R is (α, δ)-compatible skew-Armendariz. As a consequence we extend and unify several known results.     

ACCP Rises to the Polynomial Ring if the Ring has Only Finitely Many Associated Primes

Abstract

We show for a commutative ring R with unity: If R satisfies the ascending chain condition on principal ideals (accp) and has only finitely many associated primes, then for any set of indeterminates X the polynomial ring R[X] also satisfies accp. Further we show that accp rises to the power series ring R[[X]] if R satisfies accp and the ascending chain condition on annihilators.     

On the Multiplicative Monoid of n×n Matrices Over Artinian Chain Rings

Let R be an Artinian chain ring with a principal maximal ideal. We investigate properties of matrices over R and give matrix representations of R-submodules of R ⁿ first, then consider Green's relations, Green's relation equivalent classes, Schützenberger groups of 𝒟-classes, principal factors, and group ?-classes of the multiplicative monoid M _n (R) of n × n matrices over R. Furthermore, we show that M _n (R) is an eventually regular semigroup and derive basic numerical information of M _n (R) when R is finite.     

Dubrovin valuation properties of skew group rings and crossed products

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G _T=<1>,where G _T is the inertial group of R.     

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].     

Semiprimeness,quasi-Baerness and prime radical of skew generalized power series rings

Let R be a ring, (S,≤) a strictly ordered monoid and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev-Neumann Laurent series rings. In this paper we obtain necessary and sufficient conditions for the skew generalized power series ring R[[S,ω]] to be a semiprime, prime, quasi-Baer, or Baer ring. Furthermore, we study the prime radical of a skew generalized power series ring R[[S,ω]]. Our results extend and unify many existing results. In particular, we obtain new theorems on (skew) group rings, Mal’cev-Neumann Laurent series rings and the ring of generalized power series.     

Free Pairs of Bicyclic Units in the Integral Group Ring of the Dihedral Group

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.