Morita Duality for the Rings of Generalized Power Series

Let A, B be associative rings with identity, and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. For any bimodule _A M _B , we show that the bimodule _{[[ AS,≤ ]]}[M ^{S ,≤}]_{[[ BS, ≤ ]]} defines a Morita duality if and only if _A M _B defines a Morita duality and A is left noetherian, B is right noetherian. As a corollary, it is shown that the ring [[A ^{S ,≤}]] of generalized power series over A has a Morita duality if and only if A is a left noetherian ring with a Morita duality induced by a bimodule _A M _B such that B is right noetherian. Received April 13, 1999, Accepted December 12, 1999     

t-closed rings of formal power series

Let A ì BA\subset B be rings. We say that A is t-closed in B, if for each a ? Aa\in A and b ? Bb\in B such that b²-ab,b³-ab² ? Ab^2-ab,b^3-ab^2\in A, then b ? Ab\in A. We present a sufficient condition for the ring A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] to be t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]]. By an example, we show that our condition is not necessary. Even though the question is still open, some important cases are treated. For example, if A ì BA\subset B is an integral extension, or if A is p-injective, then A[[X₁,?,X_n]]A[[X_1,\ldots ,X_n]] is t-closed in B[[X₁,?,X_n]]B[[X_1,\ldots ,X_n]] if and only if A is t-closed in B.     

广义幂级数环的Morita对偶

设A,B是有单位元的环, (S,≤)是有限生成的Artin的严格全序幺半群, AMB是双模.本文证明了双模[[AS,≤]][MS,≤][[BS,≤]]定义一个Morita对偶当且仅当 AMB定义一个Morita对偶且A是左noether的,B是右noether的.因此A上的广 义幂级数环[[AS,≤]]具有Morita对偶当且仅当A是左noether的且具有由双模AMB 诱导的Morita对偶,使得B是右noether的.     

PP-Rings of Generalized Power Series

Abstract As a generalization of power series rings, Ribenboim introduced the notion of the rings of generalized power series. Let R be a commutative ring, and (S, ≤) a strictly totally ordered monoid. We prove that (1) the ring [[R ^S,≤]] of generalized power series is a PP-ring if and only if R is a PP-ring and every S-indexed subset C of B(R) (the set of all idempotents of R) has a least upper bound in B(R) and (2) if (S, ≤) also satisfies the condition that 0 ≤s for any s∈S, then the ring [[R ^S,≤ ]] is weakly PP if and only if R is weakly PP. Research supported by National Natural Science Foundation of China, 19501007, and Natural Science Foundation of Gansu, ZQ-96-01     

Bimodule herds

The notion of a bimodule herd is introduced and studied. A bimodule herd consists of a B-A bimodule, its formal dual, called a pen, and a map, called a shepherd, which satisfies unitality and coassociativity conditions. It is shown that every bimodule herd gives rise to a pair of corings and coactions. If, in addition, a bimodule herd is tame i.e. it is faithfully flat and a progenerator, or if it is a progenerator and the underlying ring extensions are split, then these corings are associated to entwining structures; the bimodule herd is a Galois comodule of these corings. The notion of a bicomodule coherd is introduced as a formal dualisation of the definition of a bimodule herd. Every bicomodule coherd defines a pair of (non-unital) rings. It is shown that a tame B-A bimodule herd defines a bicomodule coherd, and sufficient conditions for the derived rings to be isomorphic to A and B are discussed. The composition of bimodule herds via the tensor product is outlined. The notion of a bimodule herd is illustrated by the example of Galois co-objects of a commutative, faithfully flat Hopf algebra.     

广义逆多项式模的Attached素理想

设R是有单位元1的结合环,(S,≤)是严格全序Artin幺半群,M_R是右R-模,Att(M_R)与Att([M~(S,≤)]_([[R~(S,≤)]]))分别表示模M_R与广义逆多项式模[M~(S,≤)]_([[R~(S,≤)]])的所有Attached素理想组成的集合.该文主要讨论了广义幂级数环[[R~(S;≤)]]广义逆多项式模[[R~(S;≤)]]的Attached素理想的相关性质,证明了在一定条件下,有Att([M~(S,≤)]_([[R~(S,≤)]])={[[PR~(S;≤)]]P∈Att(M_R)}.这一结论表明广义逆多项式模([M~(S,≤)]_([[R~(S,≤)]])的Attached素理想在一定条件下可以用模M_R的Attached素理想来刻画,推广了Annin S在文献[1]中关于斜多项式环上逆多项式模的Attached素理想的相关结论.     

Special Properties of Rings of Generalized Power Series

Abstract

Let R be a ring and (S, ≤) a strictly ordered monoid. Properties of the ring [[R ^S,≤]] of generalized power series with coefficients in R and exponents in S are considered in this paper. It is shown that [[R ^S,≤]] is reduced (2-primal, Dedekind finite, clean, uniquely clean) if and only if R is reduced (2-primal, Dedekind finite, clean, uniquely clean, respectively) under some additional conditions. Also a necessary and sufficient condition is given for rings under which the ring [[R ^S,≤]] is a reduced left PP-ring.     

Principally Quasi-Baer skew Generalized Power Series modules

Let R be a ring and S a strictly totally ordered monoid. Let ω: S → End(R) be a monoid homomorphism. Let M _R be an ω-compatible module and either R satisfies the ascending chain conditions (ACC) on left annihilator ideals or every S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join. We show that M _R is p.q.-Baer if and only if the generalized power series module M[[S]] _{R[[S, ω]]} is p.q.-Baer. As a consequence, we deduce that for an ω-compatible ring R, the skew generalized power series ring R[[S, ω]] is right p.q.-Baer if and only if R is right p.q.-Baer and either R satisfies the ACC on left annihilator ideals or any S-indexed subset of right semicentral idempotents in R has a generalized S-indexed join in R. Examples to illustrate and delimit the theory are provided.     

Formally smooth bimodules

The notion of a formally smooth bimodule is introduced and its basic properties are analyzed. In particular it is proven that a B-A bimodule M which is a generator left B-module is formally smooth if and only if the M-Hochschild dimension of B is at most one. It is also shown that modules M which are generators in the category σ[M] of M-subgenerated modules provide natural examples of formally smooth bimodules.     

Quasi-hopf algebra actions and smash products

A classical result of Magnus asserts that a free group F has a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group F^A has a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that F^A is faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and F^A is a free A-groupp of Lyndon' type, then F^A has a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then F^A has a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, F^A has a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if F^Q has a faithful Magnus representation in Q[[Y]].     

Quotient finite dimensional modules with acc on subdirectly irreducible submodules are noetherian

A right R-module M is (Goldie) finite dimensional (= f.d.) if M contains no infinite direct sums of submodules.M is quotient f.d. (= q.f.d.) if M/K is f.d. for all submodules K.A submodule I of M is subdirectly irreducible (= SDI) if V is the intersection of all submodules S _α of M that properly contain I, then V ≠ I, equivalentlyM/I has simple essential socle V/I. A theorem of Shock [74] states that a q.f.d. right module M is Noether-ian iff every proper submodule of M is contained in a maximal submodule. Camillo [77], proved a companion theorem: M is q.f.d. iff every submodule A ≠ 0 contains a finitely generated (= f.g) submodule S such that A/S has no maximal submodules. Using these two results, and an idea of Camillo [75], we prove the theorem stated in the title.     

Invariant Constructions of Simple and Maximal Sets

The main results of the present paper are the following theorems: 1. There is no e ∈ ω such that for any A, B ? ω, S^A = W is simple in A, and if A′ ?_T B′, then S^A =* S^B. 2 There is an e ∈ ω such that for any A, B ? ω, M^A = W_e is incomplete maximal in A, and if A =* B, then M^A ?_T M^B.     

Duality and normal parts of operator modules

For an operator bimodule X over von Neumann algebras A⊆B(H) and B⊆B(K), the space of all completely bounded A,B-bimodule maps from X into B(K,H), is the bimodule dual of X. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To X a normal operator bimodule X_n is associated so that completely bounded A,B-bimodule maps from X into normal operator bimodules factorize uniquely through X_n. A construction of X_n in terms of biduals of X, A and B is presented. Various operator bimodule structures are considered on a Banach bimodule admitting a normal such structure.     

On Clean Rings

A ring R is called clean if every element of R is the sum of an idempotent and a unit. Let M be a R-module. It is obtained in this article that the endomorphism ring End(M) is clean if and only if, whenever A = M′ ⊕ B = A₁ ⊕ A₂ with M′ ? M, there is a decomposition M′ =M₁ ⊕ M₂ such that A = M′ ⊕ [A₁ ∩ (M₁ ⊕ B)] ⊕ [A₂ ∩ (M₂ ⊕ B)]. Then unit-regular endomorphism rings are also described by direct decompositions.     

Localization of Monoid Objects and Hochschild Homology

Let A be a (not necessarily commutative) monoid object in an abelian symmetric monoidal category (C, ?,1) satisfying certain conditions. In this paper, we continue our study of the localization M _S of any A-module M with respect to a subset S ? Hom _A?Bimod (A, A) that is closed under composition. In particular, we prove the following theorem: if P is an A-bimodule such that P is symmetric as a bimodule over the center Z(A) of A, we have isomorphisms HH _*(A, P) _S  ? HH _*(A, P _S ) ? HH _*(A _S , P _S ) of Hochschild homology groups.     

Projective MV-algebras and rational polyhedra

Let M be an n-generator projective MV-algebra. Then there is a rational polyhedron P in the n-cube [0, 1] ⁿ such that M is isomorphic to the MV-algebra M(P){{\rm{\mathcal {M}}}(P)} of restrictions to P of the McNaughton functions of the free n-generator MV-algebra. P necessarily contains a vertex vP of the n-cube. We characterize those polyhedra contained in the n-cube such that M(P){{\mathcal {M}}(P)} is projective. In particular, if the rational polyhedron P is a union of segments originating at some fixed vertex vP of the n-cube, then M(P){{\mathcal {M}}(P)} is projective. Using this result, we prove that if A = M(P){A = {\mathcal {M}}(P)} and B = M(Q){B = {\mathcal {M}}(Q)} are projective, then so is the subalgebra of A × B given by {(f, g) | f(v _P ) = g(v _Q ), and so is the free product A \coprod B{A \coprod B} .     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART I): A field-theoretic treatment of variational problems in n independent variables {x^j} and N dependent variables {ψ^A)} is presented that differs substantially from the standard field theories, such as those of Carathéodory [4] and Weyl [10], inasmuch as it is not stipulated ab initio that the Lagrangian be everywhere positive. This is accomplished by the systematic use of a canonical formalism. Since the latter must necessarily be prescribed by appropriate Legendre transformations, the construction of such transformations is the central theme of Part I.—The underlying manifold is M = M_n x M_N, where M_n, M_N are manifolds with local coordinates {x^j}, {ψ^A}, respectively. The basic ingredient of the theory consists of a pair of complementary distributions D_n, D_N on M that are defined respectively by the characteristic subspaces in the tangent spaces of M of two sets of smooth 1-forms {π^A:A = 1,…, N}, {π^j = 1,…, n} on M. For a given local coordinate system on M the planes of 4, have unique (adapted) basis elements B_j = (?/?_x ^j) + B^A _j (?/?ψ^A), whose coefficients B^A _j will assume the role of derivatives such as ?ψ^A/?x^j in the final analysis of Part II. The first step toward a Legendre transformation is a stipulation that prescribes B^A _j as a function of the components {π^j _h,π^j _A} of {π^j}—these components being ultimately the canonical Variables—in such a manner that B^A _j is unaffected by the action of any unimodular transformation applied to the exterior system {π^j}. A function H of the canonical variables is said to be an acceptable Hamiltonian if it satisfies a similar invariance requirement, together with a determinantal condition that involves its Hessian with respect to π^j _A. The second part of the Legendre transformation consists of the identification in terms of H and the canonical variables of a function L that depends solely on B^A _j and the coordinates on M. This identification imposes a condition on the Hessian of L with respect to B^A _j. Conversely, any function L that satisfies these requirements is an acceptable Lagrangian, whose Hamiltonian is uniquely determined by the general construction.     

On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.     

Artinness of Generalized Macaulay–Northcott Modules

Let R be an associated ring not necessarily with identity, M a left R-module having the property (F), and (S, ≤) a strictly totally ordered monoid which is also artinian and finitely generated. It is shown that the module [M ^S,≤] consisting of generalized inverse polynomials over M is an artinian left [[R ^S,≤]]-module if and only if M is an artinian left R-module.     

Dimension de l’anneau des polynomes a valeurs entieres

Let A be a commutative domain with quotient field K and A_S the ring of integer-valued polynomials thus A_S={f∈K[X]; f(A)⊂A}; we show that the Krull dimension of A_S is such that dim A_S≥dim A[X]-1 and give examples where dim A_S=dim A[X]-1. Considering chains of primes of A_S above a maximal idealm of finite residue field, we give also examples where the length of such a chain can arbitrarily be prescribed (whereas in A[X] the length of such chains is always 1). To provide such examples we consider a pair of domains A⊂B sharing an ideal I such that A/I is finite; we give sufficient conditients to have A_S⊂B[X] and show that in this case dim A_S=dim B[X]. At last, as a generalisation of Noetherian rings of dimension 1, we consider domains with an ideal I such that A/I is finite and a power Iⁿ of I is contained in a proper principal ideal of A; for such domains we show that every prime of A_S above a primem containing I is maximal.