Skew Power Series Research Articles

Completions of quantum coordinate rings

Given an iterated skew polynomial ring C[y 1 ; τ 1 , δ 1 ]...[y n ;τ n ,δ n ] over a complete local ring C with maximal ideal m, we prove, under suitable assumptions, that the completion at the ideal m+(y 1 , y 2 ;.., y n ) is an iterated skew power series ring. Under further conditions, the completion becomes a local, noetherian, Auslander regular domain. Applicable examples include quantum matrices, quantum symplectic spaces, and quantum Euclidean spaces.

Annihilator Conditions on Ore Extensions

In this paper, we extend the study of various annihilator conditions on the nearring of polynomials and the nearring of formal power series to skew polynomials and skew formal power series in which addition and substitution are used as operations. A result of Hong et al. on the skew polynomial rings and skew power series of an α-rigid Baer ring is extended to Baer conditions in the nearrings of skew polynomials and skew formal power series. Also, for an injective endomorphism α of a ring R, it is shown that R is α-rigid if and only if the nearring (R[x; α], +, ◦) is reduced, if and only if the nearring (R0[[x; α]], +, ◦) is reduced, if and only if R is a reduced and near Armendariz ring.

GENERALIZED SEMI COMMUTATIVE RINGS AND THEIR EXTENSIONS

For an endomorphism <TEX>${\alpha}$</TEX> of a ring R, the endomorphism <TEX>${\alpha}$</TEX> is called semicommutative if ab=0 implies <TEX>$aR{\alpha}(b)$</TEX>=0 for a <TEX>${\in}$</TEX> R. A ring R is called <TEX>${\alpha}$</TEX>-semicommutative if there exists a semicommutative endomorphism <TEX>${\alpha}$</TEX> of R. In this paper, various results of semicommutative rings are extended to <TEX>${\alpha}$</TEX>-semicommutative rings. In addition, we introduce the notion of an <TEX>${\alpha}$</TEX>-skew power series Armendariz ring which is an extension of Armendariz property in a ring R by considering the polynomials in the skew power series ring <TEX>$R[[x;\;{\alpha}]]$</TEX>. We show that a number of interesting properties of a ring R transfer to its the skew power series ring <TEX>$R[[x;\;{\alpha}]]$</TEX> and vice-versa such as the Baer property and the p.p.-property, when R is <TEX>${\alpha}$</TEX>-skew power series Armendariz. Several known results relating to <TEX>${\alpha}$</TEX>-rigid rings can be obtained as corollaries of our results.

ON ANNIHILATOR IDEALS OF A NEARRING OF SKEW POLYNOMIALS OVER A RING

For a ring endomorphism α and an α-derivation δ of a ring R, we study relation between the set of annihilators in R and the set of annihilators in nearring R[x; α, δ] and R0[[x; α]]. Also we extend results of Armendariz on the Baer and p.p. conditions in a polynomial ring to certain analogous annihilator conditions in a nearring of skew polynomials. These results are somewhat surprising since, in contrast to the skew polynomial ring and skew power series case, the nearring of skew polynomials and skew power series have substitution for its “multiplication”

On the codimension of modules over skew power series rings with applications to Iwasawa algebras

The first purpose of this paper is to set up a general notion of skew power series rings S over a coefficient ring R, which are then studied by filtered ring techniques. The second goal is the investigation of the class of S-modules which are finitely generated as R-modules. In the case that S and R are Auslander regular we show in particular that the codimension of M as S-module is one higher than the codimension of M as R-module. Furthermore its class in the Grothendieck group of S-modules of codimension at most one less vanishes, which is in the spirit of the Gersten conjecture for commutative regular local rings. Finally we apply these results to Iwasawa algebras of p-adic Lie groups.

Polynomial extensions of quasi-Baer rings

For a ring endomorphism α and an α-derivation δ, we introduce α-compatible rings which are a generalization of α-rigid rings, and study on the relationship between the quasi Baerness and p.q.-Baer property of a ring R and those of the polynomial extensions (including formal skew power series, skew Laurent polynomials and skew Laurent series). As a consequence we obtain a generalization of [6], [8] and [16].

SKEW POWER SERIES EXTENSIONS OF α-RIGID P.P.-RINGS

We investigate skew power series of -rigid p.p.-rings, where is an endomorphism of a ring R which is not assumed to be surjective. For an -rigid ring R, R[[]] is right p.p., if and only if R[[]] is right p.p., if and only if R is right p.p. and any countable family of idempotents in R has a join in I(R).

NILRADICALS OF SKEW POWER SERIES RINGS

For a ring endomorphism of a ring R, J. Krempa called a rigid endomorphism if a(a)＝0 implies a＝0 for a R. A ring R is called rigid if there exists a rigid endomorphism of R. In this paper, we extend the (J'-rigid property of a ring R to the upper nilradical (R) of R. For an endomorphism R and the upper nilradical (R) of a ring R, we introduce the condition (*): (R) is a -ideal of R and a(a) (R) implies a (R) for a R. We study characterizations of a ring R with an endomorphism satisfying the condition (*), and we investigate their related properties. The connections between the upper nilradical of R and the upper nilradical of the skew power series ring R[[;]] of R are also investigated.ated.

POLYNOMIAL AND SERIES RINGS AND PRINCIPAL IDEALS

All rings are assumed to be associative and with nonzero identity element. Expressions such as “a Noetherian ring” mean that the corresponding right and left conditions hold. Let A be a ring, and let φ be an injective endomorphism of A. We denote by Ar[[x, φ]] the right skew power series ring consisting of formal series ∑∞ i=0 x ai of an indeterminate x with canonical coefficients ai ∈ A, where addition is defined naturally and multiplication is defined with the use of the relation axi = xiφi(a). We denote by Ar[x, φ] the right skew polynomial ring that is a subring in Ar[[x, φ]] and consists of finite formal sums ∑ xai. We similarly define the left skew power series ring A [[x, φ]] and the left skew polynomial ring A [x, φ], where xia = φi(a)xi. A module is called a Bezout module if all its finitely generated submodules are cyclic modules. The presented work is a review of some results on skew polynomial or series rings that are Bezout rings or principal right ideal rings.

Ore extensions of Baer and p.p.-rings

We investigate Ore extensions of Baer rings and p.p.-rings. Let α be an endomorphism and δ an α-derivation of a ring R. Assume that R is an α-rigid ring. Then (1) R is a Baer ring if and only if the Ore extension R[ x; α, δ] is a Baer ring if and only if the skew power series ring R[[ x; α]] is a Baer ring, (2) R is a p.p.-ring if and only if the Ore extension R[ x; α, δ] is a p.p.-ring.

Undecidable wreath products and skew power series fields

We prove the undecidability of a very large class of restricted and unrestricted wreath products (Theorem 1.2), and of some skew fields of power series (Section2). Both undecidabilities are obtained by interpreting some enrichments of twisted wreath products, which are themselves proved to be undecidable (Proposition 1.1).We consider division rings of power series in various languages:We show (Theorem 2.8) that every power series division ring k((B)), whose field of constants k is commutative and whose ordered group of exponents is noncommutative with a convex center, is undecidable in every extension of the language of rings where the valuation and the ordered group B are definable.For certain k and B we prove here the undecidability of the structurewhere X↾k((B))xB is the restriction of the multiplication to k((B)) Χ B,and γ is a given conjugation of k((B)). This shows that we cannot hope to improve our previous result, a sort of Ax-Kochen-Ershov principle for power series division rings, which ensures thatis decidable for every decidable solvable B.

The extended center of a skew power series ring

For a prime ring R with σ ∊ Aut(R), we examine the extended center of the skew power series ring R[[x;σ]]. We prove the extended center of R[[x;σ]] is isomorphic to: (1) Cσ if no power of σ is X-inner where Cσ is the fixed field of a acting on the extended center C of R (2) Cσ((vx')) (Laurent series) if some positive power of a is X-inner and R is closed prime. In this case I is the least positive integer such that a1 is X-inner with σ1= inn(v). We provide an example showing this result fails if R is not closed prime.

Skew power series rings with general commutation formula

The noncommutative product in a skew power series ring Λ in an indeterminate X with coefficients in a skewfield K is entirely determined from a commutation law between X and any element of K. Different types of commutation rules are described, with some resulting properties for Λ and its skewfield of fractions.

Skew power series rings and derivations

369 Condition (i) follows easily from the fact that (a + b)z = az + bz and that for a # 0 the order of az equals 1. Condition (ii,) follows from the associative law (ab)z = a(bz), see also 12, 51. This means, in particular, that 6, is a monomorphism and that 6,6;’ is a Jo-derivation of F. We will say that a sequence (a,, 6, ,..., 6, ,...) of mappings di from F’to F is admissible if (i) and (ii,) hold for all M. Such a sequence could be finite. We want to investigate the set of all admissible sequences for a given field F and we assume from now on that F is commutative. One known admissible sequence is (id, 6, 6*, 6’,..., a”,...) for an ordinary derivation 6 of F. It will be proved that (id, 0,O ,..., 0, 6 = 6,. 0, 0 ,..., 0, ((k + 1)/2) 6’ = 6,,, 0 ,..., 0, 0 ,..., 0, ((2k + I)(k + 1)/(2 x 3)) 6” = 6,, ,...) is admissible for a derivation 6 of F. This sequence is just a special case of a general type of sequence discussed in this paper in which the 6; = gi(S) for a derivation 6 of F and where the gi(x) are certain polynomials in one variable x with coef- ficients in K = {a E F; us = 0). Even though it seems likely that these sequences are admisssible we were not able to prove this in general. Further, it is shown that finite admissible sequences do exist which cannot form the beginning segment of a longer admissible sequence. Another obvious problem, not dealt with here, is the investigation of equivalence classes of admissible sequences under the equivalence relation given by the isomorphism of the corresponding rings F[ [z, a,, 6, ,..., d,,... 11 with multiplication defined by (M). 1.

Skew polynomial rings and skew power series rings

Skew polynomial rings and skew power series rings

Polynomes de ore, series formelles tordues et anneaux filters complets herediataires

The Ore extension A [t ; τ, δ] of a semi-simple artinian ring A ,with , is right hereditary, even if the ring monomorphism τ is not surj ective (th . 2.1) . If δ = 0, then R is the associated graded ring of the skew power series ring , so we are led to study fil tered modules over a complete filtered ring A , with decreasing filtrations, which are projective, with respect of a certain family of epimorphisms , in the category of filtered A-modules, which we call f-projective (§§3,4). The skew power series ring over a semisimple artinian ring is right f-hereditary, i.e. every right ideal is f-projective (th. 4.5).