Formal power series representations of free exponential groups

Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications. As they are also of paramount importance in Algebraic Geometry, polynomial rings have been intensively studied. On the other hand, rings of formal power series have been extensively used in “algebroid geometry” and have many properties which are parallel to those of polynomial rings. In the first section of this chapter we shall define formal power series rings and we shall show that the main properties of polynomial rings which have been derived in previous chapters (see, in particular, Vol. I, Ch. I, §§ 16–18) hold also for formal power series rings. In the later sections of this chapter we shall give deeper properties of polynomial rings and, whenever possible, the parallel properties of power series rings.

We investigate the translation equation (T) , and the cocycle equations (Co1) , (Co2) in [ x ], the ring of formal power series over . Here we restrict ourselves to iteration groups of type I, i. e. to solutions of (T) of the form , where c 1 ? 1 is a generalized exponential function. It is easy to prove that the coefficient functions c n (s), a n (s), s n (s) of are polynomials, c 1(s), polynomials or rational functions in one more generalized exponential function and, in a special case, polynomials in an additive function. For example, we obtain . For this we do not need any detailed information about the polynomials P n . Under some conditions on the exponential and additive functions it is possible to replace the exponential and additive functions by independent indeterminates. In this way we obtain formal versions of the translation equation and the cocycle equations in rings of the form [ x ], [ x ], [ x ], [ x ], and [ x ]. We solve these equations in a completely algebraic way, by deriving formal differential equations or Aczel�Jabotinsky type equations for these problems. It is possible to get the structure of the coefficients in great detail which are now polynomials or rational functions. We prove the universal character (depending on certain parameters) of these polynomials or rational functions. And we deduce the canonical form S -1(yS(x)) for iteration groups of type I. This approach seems to be more clear and more general than the original one. Some simple substitutions allow us to solve these problems in rings of the form ( [ u ]) [ x ], i. e. where the coefficient functions are formal series.

Let P = P(z,d/dz) be a linear ordinary differential operator with holomorphic coefficients in a neighbourgood X of the origin 0 in ℂ. Let Pu = f be an inhomogeneous differential equation in the space O {0} or in O {0}, where 0 {0} is the ring of convergent power series and O {0} is the ring of formal power series at the origin 0.

Self-Dual Permutation Codes Over Formal Power Series Rings and Finite Principal Ideal Rings

Strongly Hopfian Rings of Formal Power Series and Power Series Armendariz Rings in Infinitely Many Variables

This paper deals with the unit groups of the endomor- phism rings of projective modules over polynomial rings and further over formal power series rings. A normal subgroup of the unit group is dened and discussed. The local splitting properties of elements of endomorphism rings of projective modules over polynomial rings are given. So, if P is a projective module over a ring A, then (id+ xEndA(x)(P(x))) is constructed. This is a normal subgroup of the unit group of endomorphism rings of a projective module over a poly- nomial ring. If (A,m) is a local ring with dim(A) ≥ µ(m), then we show that (id+ xEndA(x)(P(x))) is a normal subgroup of SLA(x)(P(x)). Also, we will show that under these conditions the similar conclusion can be drawn for the ring of formal power series. In section 3, let P be a projective A-module. Let s1, s2 ∈ A be such that As1 + As2 = A. Then we use the splitting lemma of Quillen to show that every element of (idPs1s2 + xEnd As1s2(x) (Ps1s2 (x))) has two decompositions. Finally, we generalize the result to Theorem 3.4.

Let R be an integral domain, Χ be a set of indeterminates over R , and R[[\mathcal X]]_3 be the full ring of formal power series in \mathcal X] over R . We show that the Picard group of R[[\mathcal X]]_3 is isomorphic to the Picard group of R . An integral domain is called a π -domain if every principal ideal is a product of prime ideals. An integral domain is a π -domain if and only if it is a Krull domain that is locally a unique factorization domain. We show that R[[\mathcal X]]_3 is a π -domain if R[[Χ_1 , . . . , Χ_n]] is a π -domain for every n ≥ 1 . In particular, R[[\mathcal X]]_3 is a π -domain if R is a Noetherian regular domain. We extend these results to rings with zero-divisors. A commutative ring R with identity is called a π -ring if every principal ideal is a product of prime ideals. We show that R[[\mathcal X]]_3 is a π -ring if R is a Noetherian regular ring.

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.

Let ℂ〚x〛 be the ring of formal power series in one indeterminate over ℂ, and let Γ be the group of automorphisms of ℂ〚x〛 which are continuous in the order topology and leave ℂ elementwise fixed. Assume that(F t ) t ∈ ℂ and(G t ) t ∈ ℂ are iteration groups, i.e. one-parameter subgroups of Γ which are solutions of the translation equationF t ∘F s =F t + s ,G t ∘G s =G t + s . Suppose moreover that the following weak commutativity condition holds: (1) $$F_t \circ G_t = G_t \circ F_t forallt \in \mathbb{C}.$$ Does (1) imply the stronger condition (2) $$F_t \circ G_s = G_s \circ F_t foralls,t \in \mathbb{C}?$$ (This problem had been posed by J. Schwaiger. Similar problems for homomorphisms of (ℂ, + ) into groups of matrices have been dealt with by Z. Moszner and Z. Leszczynska.) We give an affirmative answer to this question by characterizing all pairs (Ft)t∈ℂ, (Gt)t∈ℂ of iteration groups which satisfy (1). For such pairs of iteration groups exactly two cases occur: We do not assume that the iteration groups under consideration are analytic. Indeed, no assumption on the regularity of the dependence on the group parameter is made.

Rings of polynomials and formal power series

Some results on local cohomology of polynomial and formal power series rings: The one dimensional case

This paper deals with a reducible sl(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture: Suppose that sl(2,C) acts on the formal power series ring via (1.1). Then I(f) = (li1) ⊕ (li2) ⊕... ⊕ (lis) modulo some one dimensional sl(2,C) representations where (li) is an irreducible sl(2,C) representation of li dimension and {li1li2,...,lis} ⊆ {l1, l2...,lr}. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively.

We obtain a bijection between the set of equivalent classes of invariant star products on a non-degenerate triangular finite dimensional Lie bialgebra over the formal power series ring and the set , working in the framework developed by Etingof–Kazhdan for the quantization of Lie bialgebras. Two of the corresponding triangular Hopf algebras over the ring are isomorphic if and only if the invariant star products defining them are equivalent. Therefore, when t = ℏ, we obtain a set of triangular Hopf quantized universal enveloping algebras which can also be seen as quantizations of the deformation algebra . Additionally, two of them are isomorphic if and only if the above invariant star products are equivalent.

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.

Given an algebraically closed field $K$ of characteristic zero, we prove the Abhyankar–Jung theorem for any excellent henselian ring whose completion is a formal power series ring $K[[z]]$. In particular, examples include the local rings which form