On skew polynomial rings over locally nilpotent rings

We will show that skew polynomial rings in several variables over locally nilpotent rings cannot contain nonzero idempotent elements. We will also prove that such rings are Brown–McCoy radical.     

Projective ideals of skew polynomial rings over HNP rings

Let R be an hereditary Noetherian prime ring (or, HNP-ring, for short), and let S?=?R[x;σ] be a skew polynomial ring over R with σ being an automorphism on R. The aim of the paper is to describe completely the structure of right projective ideals of R[x;σ] where R is an HNP-ring and to obtain that any right projective ideal of S is of the form X𝔟[x;σ], where X is an invertible ideal of S and 𝔟 is a σ-invariant eventually idempotent ideal of R.     

Baer环和拟-Baer环的多项式扩张的一点注记

I1和I2分别是环R的一个左理想和右理想,T1=R[x]和T2=R[x,x-1]分别表示多项式环和洛朗多项式环.首先给出两个例子,分别说明了T1I1不一定是T1的左理想与T2L2不一定是T2的右理想.其次给出了环的多项式扩张及洛朗扩张的理想的性质.最后证明了,若R[X](R[x,x-1])是拟-Baer环,则R也是拟-...     

Left APP differential polynomial rings

A ring R is called a left APP-ring if for each element a∈R, the left annihilator l_R(Ra) is right s-unital as an ideal of R or equivalently R∕l_R(Ra) is flat as a left R-module. In this paper, we show that for a ring R and derivation δ of R, R is left APP if and only if R is δ-weakly rigid and the differential polynomial ring R[x;δ] is left APP. As a consequence, we see that if R is a left APP-ring, then the n^th Weyl algebra over R is left APP. Also we define δ-left APP (resp. p.q.-Baer) rings and we show that R is left APP (resp. p.q.-Baer) if and only if for each derivation δ of R, R is δ-weakly rigid and δ-left APP (resp. p.q.-Baer). Finally we prove that R[x;δ] is left APP (resp. p.q.-Baer) if and only if R is δ-left APP (resp. p.q.-Baer).     

Structure theorems over polynomial rings

Given a polynomial ring R over a field k and a finite group G, we consider a finitely generated graded RG-module S. We regard S as a kG-module and show that various conditions on S are equivalent, such as only containing finitely many isomorphism classes of indecomposable summands or satisfying a structure theorem in the sense of [D. Karagueuzian, P. Symonds, The module structure of a group action on a polynomial ring: A finiteness theorem, preprint, http://www.ma.umist.ac.uk/pas/preprints].     

Modules over serial rings

This paper continues the study of Noetherian serial rings. General theorems describing the structure of such rings are proved. In particular, some results concerning π-projective and π-injective modules over serial rings are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 880–892 June, 1999.     

Zero commutativity of nilpotent elements skewed by ring endomorphisms

The reversible property is an important role in noncommutative ring theory. Recently, the study of the reversible ring property on nilpotent elements is established by Abdul-Jabbar et al., introducing the concept of commutativity of nilpotent elements at zero (simply, a CNZ ring) as a generalization of reversible rings. We here study this property skewed by a ring endomorphism α, and such ring is called a right α-skew CNZ ring which is an extension of CNZ rings as well as a generalization of right α-skew reversible rings, and then investigate the structure of right α-skew CNZ rings and their related properties. Consequently, several known results are obtained as corollaries of our results.     

A Koszul complex over skew polynomial rings

We construct a Koszul complex in the category of left skew polynomial rings associated with a flat endomorphism that provides a finite free resolution of an ideal generated by a Koszul regular sequence.     

Commutativity of rings with polynomial constraints

Let p, q and r be fixed non-negative integers. In this note, it is shown that if R is left (right) s-unital ring satisfying , respectively) where , then R is commutative. Moreover, commutativity of R is also obtained under different sets of constraints on integral exponents. Also, we provide some counterexamples which show that the hypotheses are not altogether superfluous. Thus, many well-known commutativity theorems become corollaries of our results.     

Congruence of symmetric matrices over local rings

Let R be a commutative, local, and principal ideal ring with maximal ideal and residue class field F. Suppose that every element of is square. Then the problem of classifying arbitrary symmetric matrices over R by congruence naturally reduces, and is actually equivalent to, the problem of classifying invertible symmetric matrices over F by congruence.     

Constants of derivations in polynomial rings over unique factorization domains

A well-known theorem, due to Nagata and Nowicki, states that the ring of constants of any -derivation of , where is a commutative field of characteristic zero, is a polynomial ring in one variable over . In this paper we give an elementary proof of this theorem and show that it remains true if we replace by any unique factorization domain of characteristic zero.

     

Descriptions of radicals of skew polynomial and skew Laurent polynomial rings

In this paper, we first characterize the Levitzki radical of a skew (Laurent) polynomial ring by the prime ideals and skewed prime ideals in the base ring. We next provide formulas for the strongly prime radical and the uniformly strongly prime radical of skew (Laurent) polynomial rings.     

On slightly P-coherent rings

A ring R is called left slightly P-coherent if C is P-injective, for every left R-module exact sequence 0→A→B→C→0 with A and B P-injective. The properties of slightly P-coherent rings and several examples are studied to show that left slightly P-coherent rings fall in between left P-coherent rings and left strongly P-coherent rings. In terms of some derived functors, some homological dimensions over these rings are investigated. As applications, some new characterizations of p.p.rings are given.     

Integer-Valued Polynomials over Matrix Rings

When D is a commutative integral domain with field of fractions K, the ring Int(D) = {f ∈ K[x] | f(D) ? D} of integer-valued polynomials over D is well-understood. This article considers the construction of integer-valued polynomials over matrix rings with entries in an integral domain. Given an integral domain D with field of fractions K, we define Int(M _n (D)): = {f ∈ M _n (K)[x] | f(M _n (D)) ? M _n (D)}. We prove that Int(M _n (D)) is a ring and investigate its structure and ideals. We also derive a generating set for Int(M _n (?)) and prove that Int(M _n (?)) is non-Noetherian.     

On polynomial rings over a ring with a selfduality

We prove that a ringR has a self duality induced by a leftR-moduleM if and only if its polynomial ringR[x] has a graded self duality induced by a graded leftR[x]-moduleM[x ⁻¹]. Supported by the Natural Science Foundation of Fujian Province (1994–1997)