Power Series Rings Research Articles

INTERPRETING ARITHMETIC IN THE FIRST-ORDER THEORY OF ADDITION AND COPRIMALITY OF POLYNOMIAL RINGS

AbstractWe study the first-order theory of polynomial rings over a GCD domain and of the ring of formal entire functions over a non-Archimedean field in the language $\{ 1, + , \bot \}$. We show that these structures interpret the first-order theory of the semi-ring of natural numbers. Moreover, this interpretation depends only on the characteristic of the original ring, and thus we obtain uniform undecidability results for these polynomial and entire functions rings of a fixed characteristic. This work enhances results of Raphael Robinson on essential undecidability of some polynomial or formal power series rings in languages that contain no symbols related to the polynomial or power series ring structure itself.

Differential Operators of Infinite Order in the Space of Formal Laurent Series and in the Ring of Power Series with Integer Coefficients

We study the Hurwitz product (convolution) in the space of formal Laurent series over an arbitrary field of zero characteristic. We obtain the convolution equation which is satisfied by the Euler series. We find the convolution representation for an arbitrary differential operator of infinite order in the space of formal Laurent series and describe translation invariant operators in this space. Using the p-adic topology in the ring of integers, we show that any differential operator of infinite order with integer coefficients is well defined as an operator from ℤ[[z]] to ℤp[[z]].

Finite determinacy of matrices and ideals

The main aim of this paper is to characterize ideals I in the power series ring R=K[[x1,…,xs]] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim(R/I)>0 and K is an infinite field (of arbitrary characteristic). Here two ideals I and J are contact equivalent if the local K–algebras R/I and R/J are isomorphic. If I is minimally generated by a1,…,am, we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b1,…,bm with ai−bi∈〈x1,…,xs〉k for some integer k. We give also computable and semicontinuous determinacy bounds.The above result is proved by considering left–right equivalence on the ring Mm,n of m×n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in Mm,n have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture.

Stillman’s conjecture via generic initial ideals

Using recent work by Erman–Sam–Snowden, we show that finitely generated ideals in the ring of bounded-degree formal power series in infinitely many variables have finitely generated Gröbner bases relative to the graded reverse lexicographic order. We then combine this result with the first author’s work on topological Noetherianity of polynomial functors to give an algorithmic proof of the following statement: ideals in polynomial rings generated by a fixed number of homogeneous polynomials of fixed degrees only have a finite number of possible generic initial ideals, independently of the number of variables that they involve and independently of the characteristic of the ground field. Our algorithm outputs not only a finite list of possible generic initial ideals, but also finite descriptions of the corresponding strata in the space of coefficients.Dedicated to Gennady Lyubeznik on the occasion of his 60th birthday.

Examples of strongly clean rings

A ring is called strongly clean if each of its elements can be written as the sum of an idempotent and a unit which commute. In this paper, we deal with two questions: first, is the center of a strongly clean ring R strongly clean, and second, is the power series ring over a strongly clean ring R strongly clean? Both questions have a positive answer when R is optimally clean, which is slightly more than strongly clean. We show that in general the questions have a negative answer, by giving, for any integral domain k, a strongly clean ring R such that and is not strongly clean. Moreover, we show that the optimally clean property is not a necessary condition for the answer to be positive, by giving, for any integral domain k, a strongly clean ring R such that and is strongly clean, but R is not optimally clean.

The symmetric signature of cyclic quotient singularities

The symmetric signature is an invariant of local domains which was recently introduced by Brenner and the first author in an attempt to find a replacement for the $F$-signature in characteristic zero. In the present note we compute the symmetric signature for two-dimensional cyclic quotient singularities, i.e. invariant subrings $k[[u,v]]^G$ of rings of formal power series under the action of a cyclic group $G$. Equivalently, these rings arise as the completions (at the irrelevant ideal) of two-dimensional normal toric rings. We show that for this class of rings the symmetric signature coincides with the $F$-signature.

Repeat-Accumulate Signal Codes

State-constrained signal codes directly encode modulation signals using signal processing filters, the coefficients of which are constrained over the rings of formal power series. Although the performance of signal codes is defined by these signal filters, optimal filters must be found by brute-force search in terms of the symbol error rate because the asymptotic behavior with different filters has not been investigated. Moreover, the computational complexity of the conventional BCJR used in the decoder increases exponentially, as the number of output constellations increases. We hence propose a new class of state-constrained signal codes called repeat-accumulate signal codes (RASCs). To analyze the asymptotic behavior of these codes, we employ Monte Carlo density evolution (MC-DE). As a result, the optimum filters can be efficiently found for the given parameters of the encoder. We also introduce a low-complexity decoding algorithm for RASCs called the extended min-sum (EMS) decoder. The MC-DE analysis shows that the difference between the noise thresholds of RASC and the Shannon limit is within 0.8 dB. The simulation results moreover show that the EMS decoder can reduce the computational complexity to less than 25% of that of conventional decoder without degrading the performance by more than 1 dB.

On S-GCD domains

Let [Formula: see text] be a multiplicative set in an integral domain [Formula: see text]. A nonzero ideal [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-[Formula: see text]-principal if there exist an [Formula: see text] and [Formula: see text] such that [Formula: see text]. Call [Formula: see text] an [Formula: see text]-GCD domain if each finitely generated nonzero ideal of [Formula: see text] is [Formula: see text]-[Formula: see text]-principal. This notion was introduced in [A. Hamed and S. Hizem, On the class group and [Formula: see text]-class group of formal power series rings, J. Pure Appl. Algebra 221 (2017) 2869–2879]. One aim of this paper is to characterize [Formula: see text]-GCD domains, giving several equivalent conditions and showing that if [Formula: see text] is an [Formula: see text]-GCD domain then [Formula: see text] is a GCD domain but not conversely. Also we prove that if [Formula: see text] is an [Formula: see text]-GCD [Formula: see text]-Noetherian domain such that every prime [Formula: see text]-ideal disjoint from [Formula: see text] is a [Formula: see text]-ideal, then [Formula: see text] is [Formula: see text]-factorial and we give an example of an [Formula: see text]-GCD [Formula: see text]-Noetherian domain which is not [Formula: see text]-factorial. We also consider polynomial and power series extensions of [Formula: see text]-GCD domains. We call [Formula: see text] a sublocally [Formula: see text]-GCD domain if [Formula: see text] is a [Formula: see text]-GCD domain for every non-unit [Formula: see text] and show, among other things, that a non-quasilocal sublocally [Formula: see text]-GCD domain is a generalized GCD domain (i.e. for all [Formula: see text] is invertible).

Anti-Archimedean property and the formal power series rings

We investigate the anti-Archimedean property in the power series rings. We characterize the bounded elements of . We then show that if D is an anti-Archimedean domain, then and are anti-Archimedean. We also show that for an integrally closed domain D, D being anti-Archimedean, anti-Archimedean, and anti-Archimedean are all equivalent.

On q-commutative power and Laurent series rings at roots of unity

We continue the first and second authors’ study of q-commutative power series rings and Laurent series rings , specializing to the case in which the commutation parameters qij are all roots of unity. In this setting, R is a PI algebra, and we can apply results of De Concini, Kac, and Procesi to show that L is an Azumaya algebra whose degree can be inferred from the qij. Our main results establish an exact criterion (dependent on the qij) for determining when the centers of L and R are commutative Laurent series and commutative power series rings, respectively. In the event this criterion is satisfied, it follows that L is a unique factorization ring in the sense of Chatters and Jordan, and it further follows, by results of Dumas, Launois, Lenagan, and Rigal, that R is a unique factorization ring.

The power series Dedekind-Mertens number

Let and be the polynomial ring and the power series ring, respectively, over a commutative ring R with identity. For , let Af be the ideal of R generated by the coefficients of f. If g is a polynomial in , then the Dedekind-Mertens number is the least positive integer k such that for every polynomial . For a power series , we define the power series Dedekind-Mertens number to be the least positive integer k (if any) such that for every power series . Heinzer and Huneke showed that if , then -, where l- is the locally minimal number of generators of Ag. Surprisingly, for , Epstein and Shapiro obtained a similar result that - provided that the ring R is Noetherian. They asked whether the same result still holds for an arbitrary ring Park, Kang, and Toan gave a negative answer to this question and further proved that if , then -, where l- is the locally minimal number of special generators of Ag. Note that we always have l-- for every power series . In this paper, we show that there is a ring R with power series such that - while l- can be arbitrarily larger than l-. In other words, the upper bound l- for is sharp and l- and are in general unrelated to the number l-. This gives a complete answer to the question by Epstein and Shapiro.

Frobenius and Cartier Algebras of Stanley–Reisner Rings (II)

It is known that the Frobenius algebra of the injective hull of the residue field of a complete Stanley–Reisner ring (i.e., a formal power series ring modulo a squarefree monomial ideal) can be only principally generated or infinitely generated as algebra over its degree zero piece, and that this fact can be read off in the corresponding simplicial complex; in the infinite case, we exhibit a 1–1 correspondence between potential new generators appearing on each graded piece and certain pairs of faces of such a simplicial complex, and we use it to provide an alternative proof of the fact that these Frobenius algebras can only be either principally generated or infinitely generated.

The lamplighter group of rank two generated by a bireversible automaton

We construct a 4-state 2-letter bireversible automaton generating the lamplighter group of rank two. The action of the generators on the boundary of the tree can be induced by the affine transformations on the ring of formal power series over .

Factorization of ideals

We study the factorization of ideals of a commutative ring, defining multiple different kinds of “nonfactorable” ideals and several “factorability” properties weaker than unique factorization. We characterize (some of) these notions, determine the implications between them, and give several examples to illustrate the differences. We also examine how these properties behave with respect to localization, direct products, idealizations, polynomial rings, monoid domains, (generalized) power series rings, and the classical D + M construction. Along the way, we give some new characterizations of the finite superideal rings introduced by A.J. Hetzel and A.M. Lawson.

Skew Generalized Power Series Rings and the McCoy Property

Given a ring $R$, a strictly totally ordered monoid $(S,\preceq)$ and a monoid homomorphism $\omega \colon S \to \operatorname{End}(R)$, one can construct the skew generalized power series ring $R[[S,\omega,\preceq]]$, consisting all of the functions from a monoid $S$ to a coefficient ring $R$ whose support is artinian and narrow, where the addition is pointwise, and the multiplication is given by convolution twisted by an action $\omega$ of the monoid $S$ on the ring $R$. In this paper, we consider the problem of determining some annihilator and zero-divisor properties of the skew generalized power series ring $R[[S,\omega,\preceq]]$ over an associative non-commutative ring $R$. Providing many examples, we investigate relations between McCoy property of skew generalized power series ring, namely $(S,\omega)$-McCoy property, and other standard ring-theoretic properties. We show that if $R$ is a local ring such that its Jacobson radical $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy. Also if $R$ is a semicommutative semiregular ring such that $J(R)$ is nilpotent, then $R$ is $(S,\omega)$-McCoy ring.

A dichotomy for the injective dimension of F-finite F-modules and holonomic D-modules

Let M be either a holonomic D-module over a formal power series ring with coefficients in a field of characteristic zero, or an F-finite F-module over a noetherian regular ring of characteristic p > 0. We prove that enjoys a dichotomy property: it has only two possible values, or .

A fresh look into monoid rings and formal power series rings

In this paper, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides canonical approaches in order to find easy and rigorous proofs and methods for many facts on polynomials and formal power series; some of them as sample are treated in this paper. Besides the universal properties of the monoid rings and polynomial rings, a universal property for the formal power series rings is also established.

Annihilators and extensions of idempotent-generated ideals

We define a ring R to be right -Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. This class of rings generalizes the class of right p.q.-Baer rings. We also define a ring R to be right I-extending if each ideal generated by an idempotent is essential, as a right R-module, in a right ideal generated by an idempotent. It is shown that the two conditions are equivalent in a semiprime ring. From the definition of a right -Baer ring, we define a generalization of a prime ring which is equivalent to the condition that all nonzero cyclic projective right R-modules are faithful. Such a ring is called right I-prime. For a semiprime ring, we show the existence of a -Baer hull. We also provide some results about the p.q.-Baer hull and when it is equal to the -Baer hull. Polynomial and formal power series rings are studied with respect to the right -Baer condition. In general, a formal power series ring over one indeterminate in which its base ring is right p.q.-Baer ring is not necessarily right p.q.-Baer. However, if the base ring is right -Baer then the formal power series ring over one indeterminate is right -Baer. The last section is devoted to matrix extensions of right -Baer rings. A characterization of when a 2-by-2 generalized upper triangular matrix ring is right -Baer is given. The last major theorem is a decomposition of a -Baer ring R, satisfying a mild finiteness condition, into a direct sum of rings where A is a direct sum of right I-prime rings; and B is a generalized triangular matrix ring with right I-prime rings down the main diagonal. Examples illustrating and delimiting our results are provided.

The Density of Ramified Primes

Let F be a number field, \mathcal{O} be a domain with fraction field \mathcal{K} of characteristic zero and \rho:\text{Gal}(\overline F/F)\to\text{GL}_n(\mathcal{O}) be a representation such that \rho\otimes\overline{\mathcal{K}} is semisimple. If \mathcal{O} admits a finite monomorphism from a power series ring with coefficients in a p -adic integer ring (resp. \mathcal{O} is an affinoid algebra over a p -adic number field) and \rho is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes of \rho is of density zero. If \mathcal{O} is a complete local Noetherian ring over \mathbb{Z}_p with finite residue field of characteristic p,\rho is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of \rho form a Zariski-dense subset of \text{Spec}\mathcal{O} , then we show that the set of ramified primes of \rho is of density zero. These results are analogues, in the context of big Galois representations, of a result of Khare and Rajan, and are proved relying on their result.

A Study of Cyclic Codes Via a Surjective Mapping

In this article, cyclic codes of length $n$ over a formal power series ring and cyclic codes of length $nl$ over a finite field are studied. By defining a module isomorphism between $R^n$ and $(Z_4)^{2^kn}$, Dinh and Lopez-Permouth proved that a cyclic shift in $(Z_4)^{2^kn}$ corresponds to a constacyclic shift in $R^n$ by $u$, where $R=\frac{Z_4[u]}{<u^{2^k}-1>}$. We have defined a bijective mapping $\Phi_l$ on $R_{\infty}$, where $R_{\infty}$ is the formal power series ring over a finite field $\mathbb{F}$. We have proved that a cyclic shift in $(\mathbb{F})^{ln}$ corresponds to a $\Phi_l-$cyclic shift in $(R_{\infty})^n$ by defining a mapping from $(R_{\infty})^n$ onto $(\mathbb{F})^{ln}$. We have also derived some related results.