Power Series Rings Research Articles

On generalized power series rings with some restrictions on zero-divisors

Let [Formula: see text] be a ring and [Formula: see text] a strictly ordered monoid. The construction of generalized power series ring [Formula: see text] generalizes some ring constructions such as polynomial rings, group rings, power series rings and Mal’cev–Neumann construction. In this paper, for a reversible right Noetherian ring [Formula: see text] and a m.a.n.u.p. monoid [Formula: see text], it is shown that (i) [Formula: see text] is power-serieswise [Formula: see text]-McCoy, (ii) [Formula: see text] have Property (A), (iii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip, (iv) [Formula: see text] is strongly AB if and only if [Formula: see text] is strongly AB. Also we study the interplay between ring-theoretical properties of a generalized power series ring [Formula: see text] and the graph-theoretical properties of its undirected zero divisor graph of [Formula: see text]. A complete characterization for the possible diameters [Formula: see text] is given exclusively in terms of the ideals of [Formula: see text]. Also, we present some examples to show that the assumption “R is right Noetherian” in our main results is not superfluous.

Krull dimension of power series rings over non-SFT domains

A ring D is called an SFT ring if for each ideal I of D, there exist a finitely generated ideal J of D with J⊆I and a positive integer k such that ak∈J for all a∈I. For a cardinal number α and a ring D, we say that dim⁡D≥α if D has a chain of prime ideals with length ≥α. Arnold showed that if D is a non-SFT ring, then dim⁡D〚X〛≥ℵ0. Let C be the class of non-SFT domains. The class C includes the class of finite-dimensional nondiscrete valuation domains, the class of non-Noetherian almost Dedekind domains, the class of completely integrally closed domains that are not Krull domains, the class of integral domains with non-Noetherian prime spectrum, and the class of integral domains with a nonzero proper idempotent ideal. The ring of algebraic integers, the ring of integer-valued polynomials on Z, and the ring of entire functions are also members of the class C. In this paper we prove that dim⁡D〚X〛≥2ℵ1 for every D∈C and that under the continuum hypothesis 2ℵ1 is the greatest lower bound of dim⁡D〚X〛 for D∈C. On the other hand, there exists a (finite-dimensional) SFT domain D such that dim⁡D〚X〛≥2ℵ1.

Koszul cycles and Golod rings

Let S be the power series ring or the polynomial ring over a field K in the variables $$x_1,\ldots ,x_n$$ , and let $$R=S/I$$ , where I is proper ideal which we assume to be graded if S is the polynomial ring. We give an explicit description of the cycles of the Koszul complex whose homology classes generate the Koszul homology of $$R=S/I$$ with respect to $$x_1,\ldots ,x_n$$ . The description is given in terms of the data of the free S-resolution of R. The result is used to determine classes of Golod ideals, among them proper ordinary powers and proper symbolic powers of monomial ideals. Our theory is also applied to stretched local rings.

Transcendental degree in power series rings

Let D be an integral domain with quotient field K. Let D[[x]] and K[[x]] be the power series ring over D and K, respectively. In this paper, we show that either (1) K[[x]] and D[[x]] have the same quotient field or (2) the quotient field of K[[x]] has uncountable transcendence degree over that of D[[x]], i.e., tr.d.(K[[x]]/D[[x]])≥ℵ1. In (2), the bound ℵ1 is the greatest lower bound that one can obtain since under the continuum hypothesis the cardinality of the quotient field of K[[x]] is exactly ℵ1 provided that K is countable. We also show that the above result holds when K is replaced by any quotient overring DS of D.

Some strange behaviors of the power series ring R[[X

LetRbe a commutative ring with identity. LetR[X]andR[[X]]be the polynomial ring and the power series ring respectively over R. Being the completion ofR[X](under the X-adic topology),R[[X]]does not always share the same property withR[X]. In this paper, we present some known strange behaviors ofR[[X]]compared to those ofR[X].

The Hurwitz Product, p-Adic Topology on ℤ, and Fundamental Solution to Linear Differential Equation in the Ring ℤ[[x

We study the linear inhomogeneous first order differential equation by′ + f(x) = y in the ring of formal power series with integer coefficients. Using the p-adic topology on the ring of integers, we construct a counterpart of the Hurwitz product of the Euler series and an arbitrary formal power series with integer coefficients. It is shown that the Euler series can be interpreted as the fundamental solution to the equation under consideration. Bibliography: 8 titles.

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a b , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a b through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.

Composition rings from formal power series rings

ABSTRACTWe study the ideals in some Dickson nearrings of polynomials and formal power series. For some of their related quotients, we introduce variants and generalizations, and construct composition rings also.

A Function Determined by a Hypersurface of Positive Characteristic

Let $R=k [[X_1, \dots ,X_{n+1}]]$ be a formal power series ring over a perfect field $k$ of characteristic $p>0$, and let $\mathfrak{m} = (X_1 , \dots , X_{n+1})$ be the maximal ideal of $R$. Suppose $0\neq f \in\mathfrak{m}$. In this paper, we introduce a function $\xi_{f}(x)$ associated with a hypersurface $R/(f)$ defined on the closed interval $[0,1]$ in $\mathbb{R}$. The Hilbert-Kunz multiplicity and the F-signature of $R/(f)$ appear as the values of our function $\xi_{f}(x)$ on the interval's endpoints. The F-signature of the pair, denoted by $s(R,f^{t})$, was defined by Blickle, Schwede and Tucker. Our function $\xi_{f}(x)$ is integrable, and the integral $\int_{t}^{1}\xi_{f}(x)dx$ is just $s(R,f^{t})$ for any $t\in[0,1]$.

Study of skew inverse Laurent series rings

In the present note, we continue the study of skew inverse Laurent series ring [Formula: see text] and skew inverse power series ring [Formula: see text], where [Formula: see text] is a ring equipped with an automorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. Necessary and sufficient conditions are obtained for [Formula: see text] to satisfy a certain ring property which is among being local, semilocal, semiperfect, semiregular, left quasi-duo, (uniquely) clean, exchange, projective-free and [Formula: see text]-ring, respectively. It is shown here that [Formula: see text] (respectively [Formula: see text]) is a domain satisfying the ascending chain condition (Acc) on principal left (respectively right) ideals if and only if so does [Formula: see text]. Also, we investigate the problem when a skew inverse Laurent series ring [Formula: see text] has the same Goldie rank as the ring [Formula: see text] and is proved that, if [Formula: see text] is a semiprime right Goldie ring, then [Formula: see text] is semiprimitive. Furthermore, we study on the relationship between the simplicity, semiprimeness, quasi-Baerness and Baerness property of a ring [Formula: see text] and these of the skew inverse Laurent series ring. Finally, we consider the problem of determining when [Formula: see text] is nilpotent.

D-modules, Bernstein–Sato polynomials and F-invariants of direct summands

We study the structure of D-modules over a ring R which is a direct summand of a polynomial or a power series ring S with coefficients over a field. We relate properties of D-modules over R to D-modules over S. We show that the localization Rf and the local cohomology module HIi(R) have finite length as D-modules over R. Furthermore, we show the existence of the Bernstein–Sato polynomial for elements in R. In positive characteristic, we use this relation between D-modules over R and S to show that the set of F-jumping numbers of an ideal I⊆R is contained in the set of F-jumping numbers of its extension in S. As a consequence, the F-jumping numbers of I in R form a discrete set of rational numbers. We also relate the Bernstein–Sato polynomial in R with the F-thresholds and the F-jumping numbers in R.

McCoy property and nilpotent elements of skew generalized power series rings

Let [Formula: see text] be a ring, [Formula: see text] a strictly ordered monoid and [Formula: see text] a monoid homomorphism. The skew generalized power series ring [Formula: see text] 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 consider the problem of determining when [Formula: see text] is nilpotent in [Formula: see text]. We study various annihilator properties and a variety of conditions and related properties that the skew generalized power series [Formula: see text] inherits from [Formula: see text]. We also introduce and study the [Formula: see text]-McCoy condition on [Formula: see text], a generalization of the standard McCoy condition from polynomials to skew generalized power series. We resolve the structure of [Formula: see text]-McCoy rings and obtain various necessary or sufficient conditions for a ring to be [Formula: see text]-McCoy. As particular cases of our general results we obtain several new theorems on the McCoy condition. Moreover various examples of [Formula: see text]-McCoy rings are provided.

Iterated socles and integral dependence in regular rings

Let R R be a formal power series ring over a field, with maximal ideal m \mathfrak {m} , and let I I be an ideal of R R . We study iterated socles of I I , that is, ideals of the form I : R m s I :_R {\mathfrak m}^s for positive integers s s . We are interested in iterated socles in connection with the notion of integral dependence of ideals. In this article we show that iterated socles are integral over I I , with reduction number at most one, provided s ≤ o ( I 1 ( φ d ) ) − 1 s \leq \text {o}(I_1(\varphi _d))-1 , where o ( I 1 ( φ d ) ) \text {o}(I_1(\varphi _d)) is the order of the ideal of entries of the last map in a minimal free R R -resolution of R / I R/I . In characteristic zero, we also provide formulas for the generators of iterated socles whenever s ≤ o ( I 1 ( φ d ) ) s\leq \text {o}(I_1(\varphi _d)) . This result generalizes previous work of Herzog, who gave formulas for the socle generators of any homogeneous ideal I I in terms of Jacobian determinants of the entries of the matrices in a minimal homogeneous free R R -resolution of R / I R/I . Applications are given to iterated socles of determinantal ideals with generic height. In particular, we give surprisingly simple formulas for iterated socles of height two ideals in a power series ring in two variables. The generators of these socles are suitable determinants obtained from the Hilbert-Burch matrix.

Dedekind–Mertens lemma and content formulas in power series rings

Let R〚X〛 be the power series ring over a commutative ring R with identity. For f∈R〚X〛, let Af denote the content ideal of f, i.e., the ideal of R generated by the coefficients of f. We show that if R is a Prüfer domain and if g∈R〚X〛 such that Ag is locally finitely generated (or equivalently locally principal), then a Dedekind–Mertens type formula holds for g, namely Af2Ag=AfAfg for all f∈R〚X〛. More generally for a Prüfer domain R, we prove the content formula (AfAg)2=(AfAg)Afg for all f,g∈R〚X〛. As a consequence it is shown that an integral domain R is completely integrally closed if and only if (AfAg)v=(Afg)v for all nonzero f,g∈R〚X〛, which is a beautiful result corresponding to the well-known fact that an integral domain R is integrally closed if and only if (AfAg)v=(Afg)v for all nonzero f,g∈R[X], where R[X] is the polynomial ring over R.For a ring R and g∈R〚X〛, if Ag is not locally finitely generated, then there may be no positive integer k such that Afk+1Ag=AfkAfg for all f∈R〚X〛. Assuming that the locally minimal number of generators of Ag is k+1, Epstein and Shapiro posed a question about the validation of the formula Afk+1Ag=AfkAfg for all f∈R〚X〛. We give a negative answer to this question and show that the finiteness of the locally minimal number of special generators of Ag is in fact a more suitable assumption. More precisely we prove that if the locally minimal number of special generators of Ag is k+1, then Afk+1Ag=AfkAfg for all f∈R〚X〛. As a consequence we show that if Ag is finitely generated (in particular if g∈R[X]), then there exists a nonnegative integer k such that Afk+1Ag=AfkAfg for all f∈R〚X〛.

On power series rings over valuation domains

ABSTRACTLet V be a valuation domain and V[[X]] be the power series ring over V. In this paper, we show that if V[[X]] is a locally finite intersection of valuation domains, then V is an SFT domain and hence a discrete valuation domain. As a consequence, it is shown that the power series ring V[[X]] is a Krull domain if and only if V[[X]] is a generalized Krull domain if and only if V[[X]] is an integral domain of Krull type (or equivalently, a PvMD of finite t-character) if and only if V is a discrete valuation domain with Krull dimension at most one.

Nil-Armendariz Condition on Skew Generalized Power Series Rings

Let R be a ring, S a strictly ordered monoid, and $$\omega :S \to {\text{End}}(R)$$ 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 Malcev–Neumann Laurent series rings. In the current work, we study the (S, ω)-nil Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials and power series to skew generalized power series. We resolve the structure of (S, ω)-nil Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S, ω)-nil Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. For example, we show that left uniserial nilpotent semicommutative rings are nil Armendariz. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and these of the ring R[[S, ω]].

Formal functional equations and generalized Lie–Gröbner series

Studying the translation equation , , for , , or the associated system of cocycle equations in rings of formal power series it is well known that the coefficient functions of their solutions are polynomials in additive and generalized exponential functions. Replacing these functions by indeterminates we obtain formal functional equations. Applying formal differentiation operators to these formal equations we obtain different types of formal differential equations. They can be solved in order to get explicit representations of the coefficient functions. In the present paper we consider iteration groups of type II, i.e. solutions of the translation equation of the form , , where and is an additive function different from 0. They correspond to formal iteration groups of type II, which turn out to be the Lie–Gröbner series . Here the Lie–Gröbner operator D is defined by for where H is the formal generator of G. Using this particular form of the formal iteration group G we are able to find short proofs and elegant representations of the solutions of the cocycle equations. In connection with the second cocycle equation we study the generalized Lie–Gröbner operator , , where are given. It yields the corresponding generalized Lie–Gröbner series which appears in the presentation of the solution of the second cocycle equation.

Nested Artin strong approximation property

We study the Artin Approximation property with constraints in a different frame. As a consequence we give a nested Artin Strong Approximation property for algebraic power series rings over a field.

English

A ring R is called a right PS-ring if its socle, Soc(R R ), is projective. Nicholson and Watters have shown that if R is a right PS-ring, then so are the polynomial ring R[x] and power series ring R[[x]]. In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R[[x −1; α, δ]] and the skew polynomial ring R[x; α, δ], where R is an associative ring equipped with an automorphism α and an α-derivation δ. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.

The group of multi-dimensional Riordan arrays

We first consider two groups, F0={g∈C[[z]]|g(0)≠0} under multiplication and F1=zF0 under composition, where C[[z]] is the ring of formal power series over the complex field. It is known that the Riordan group R is isomorphic to the semidirect product F0⋊F1. It may be viewed as a group extension of F1 by F0. In this paper, the group of three-dimensional Riordan arrays is obtained from an extension of the group R by F0. This concept extends to the group of multi-dimensional Riordan arrays. As an application, we illustrate the use of the three-dimensional Riordan array in multiple combinatorial sums.