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.     

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     

The Ascending Chain Condition for Principal Ideals of Rings of Generalized Power Series

Abstract

Let (S, ≤) be a strictly totally ordered monoid and R a domain. It is shown in this paper that [[R ^S,≤]], the ring of generalized power series with coefficients in R and exponents in S, satisfies the ascending chain condition for principal ideals if and only if the ring R and the monoid S satisfy the ascending chain condition for principal ideals of R, and of S, respectively.     

Co-Hopfian Modules of Generalized Inverse Polynomials

Let R be an associative ring not necessarily possessing an identity and (S, ≤) a strictly totally ordered monoid which is also artinian and satisfies that 0 ≤s for any s∈S. Assume that M is a left R-module having propertiy (F). It is shown that M is a co-Hopfian left R-module if and only if [M ^{S , ≤}] is a co-Hopfian left [[R ^{S , ≤}]]-module. Received October 14, 1998, Accepted October 15, 1999     

Transfer of Krull Dimension,Lying-Over,and Going-Down to the Fixed Ring

Let G be a group acting via ring automorphisms on a commutative unital ring R. If Spec(R) has no infinite antichains and either R a domain or G finitely generated, then R ^G  ? R has the lying-over property. If R is semiquasilocal and dim(R) = 0, then dim(R ^G ) = 0. If 1 ≤ d ≤ ∞, new examples are given such that d = dim(R) ≠ dim(R ^G ) < ∞. If G is locally finite on R, then R ^G  ? R satisfies universally going-down. Consequently, if G is locally finite, the S-domain, strong S-domain and universally strong S-domain properties descend from R to R ^G . If R is a domain, then G is locally finite on R ? R is integral over R ^G . One cannot delete the “domain” hypothesis.     

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.     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

On the genus of generalized unit and unitary Cayley graphs of a commutative ring

Let R be a commutative ring, U(R) be the set of all unit elements of R, G be a multiplicative subgroup of U(R) and S be a non-empty subset of G such that S ^?1={s ^?1:?s∈S}?S. In [16], K. Khashyarmanesh et al. defined a graph of R, denoted by Γ(R,G,S), which generalizes both unit and unitary Cayley graphs of R. In this paper, we derive several bounds for the genus of Γ(R,U(R),S). Moreover, we characterize all commutative Artinian rings R for which the genus of Γ(R,U(R),S) is one. This leads to the characterization of all commutative Artinian rings whose unit and unitary Cayley graphs have genus one.     

On defining sets of vertices of the hypercube by linear inequalities

This paper shows that for any subset S of vertices of the n-dimensional hypercube, ind(S)≤2^n?1, where ind(S) is the minimum number of linear inequalities needed to define S. Furthermore, for any k in the range 1≤k≤2^n?1, there is an S with ind(S) = k, with the defining inequalities taken as canonical cuts. Other related results are included, and all are proven by explicit constructions of the sets S or explicit definitions of such sets by linear inequalities.The paper is aimed at researchers in bivalent programming, since it provides upper bounds on the performance of algorithms which combine several linear constraints into one, even when the given constraints have a particularly simple form.     

On t-closedness of generalized power series rings

For an extension A⊆B of commutative rings, we present a sufficient condition for the ring [[A^S,?]] of generalized power series to be t-closed in [[B^S,?]], where (S,?) is a torsion-free cancellative ordered monoid. As a corollary, this result can be applied to the ring of power series in any number of indeterminates.     

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.     

The total graph and regular graph of a commutative ring

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z(R) is not an ideal. In this paper we show that if T(Γ(R)) is a connected graph, then . Also, we prove that if R is a finite ring, then T(Γ(R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.     

On directed zero-divisor graphs of finite rings

For an artinian ring R, the directed zero-divisor graph Γ(R) is connected if and only if there is no proper one-sided identity element in R. Sinks and sources are characterized and clarified for a finite ring R. Especially, it is proved that for any ring R, if there exists a source y in Γ(R) with y²=0, then |R|=4 and R={0,x,y,z}, where x and z are left identity elements and yx=0=yz. Such a ring R is also the only ring such that Γ(R) has exactly one source. This shows that Γ(R) cannot be a network for any finite or infinite ring R.     

Skew linear recurring sequences of maximal period over galois rings

Let p be a prime number, and R = GR(q ^d , p ^d ) be a Galois ring of q ^d = p ^rd elements and of characteristic p ^d . Denote by S = GR(q ^nd , p ^d ) a Galois extension of the ring R of dimension n and by ? the ring of all linear transformations of the module _R S. We call a sequence v over the ring S with the law of recursion $$ {\mathrm{for}\ \mathrm{all}\ }i \in {\mathbb{N}_0}:v\left( {i + m} \right) = {\psi_{m - 1}}\left( {v\left( {i + m - 1} \right)} \right) + \cdots + {\psi_0}\left( {v(i)} \right),\quad {\psi_0}, \ldots, {\psi_{m - 1}} \in \textit{\v{S}} $$ (i.e., a linear recurring sequence of order m over the module _? S) a skew LRS over S. It is known that the period T(v) of such a sequence satisfies the inequality T(v) ?? ?? = (q ^nm ?1)p ^d?1. If T(v) = ?? , then we call v a skew LRS of maximal period (a skew MP LRS) over S. A new general characterization of skew MP LRS in terms of coordinate sequences corresponding to some basis of a free module _R S is given. A simple constructive method of building a big enough class of skew MP LRS is stated, and it is proved that the linear complexity of some of them (the rank of the linear recurring sequence) over the module _S S is equal to mn, i.e., to the linear complexity over the module _R S.     

On the lattice of matric-extensible radicals

A radical α in the universal class of associative rings is called matric-extensible if α (R _n) = (α (R))_n for any ring R, and natural number n, where R _n denotes the nxn matrix ring with entries from R. We investigate matric-extensibility of the lower radical determined by a simple ring S. This enables us to find necessary and sufficient conditions for the lower radical determined by S to be an atom in the lattice of hereditary matric-extensible radicals. We also show that this lattice has atoms which are not of this form. We then describe all atoms of the lattice, and show that it is atomic.     

Infinitely many Gerretsen-Blundon style quadratic inequalities, all strongest in Blundon’s sense

Blundon has proved that if R, r and s are respectively the circumradius, the inradius and the semiperimeter of a triangle, then the strongest possible inequalities of the form q(R, r) ≤ s ² ≤ Q(R, r) that hold for all triangles becoming equalities for the equilaterals where q, Q real quadratic forms, occur for the Gerretsen forms q _B (R, r) = 16Rr ? 5r ² and Q _B (R, r) = 4R ² + 4Rr + 3r ²; strongest in the sense that if Q is a quadratic form and s ² ≤ Q(R, r) ≤ Q _B (R, r) for all triangles then Q(R, r) = Q _B (R, r), and similarly for q _B (R, r). In this paper we prove that Q _B (resp. q _B ) is just one of infinitely many forms that appear as minimal (resp. maximal) elements in the partial order induced by the comparability relation in a certain set of forms, and we conclude that all these minimal forms are strongest in Blundon’s sense. We actually find all possible such strongest forms. Moreover we find all possible quadratic forms q, Q for which q(R, r) ≤ s ² ≤ Q(R, r) for all triangles and which hold as equalities for the equilaterals.     

Injektive Moduln über Ringen linearer Differentialoperatoren

LetR be a division ring of characteristic 0 withn commuting (partial) differentiationsd _i . DefineR[d]=R[d ₁, ...,d _n ] to be all polynomials ind _i with coefficients inR. A typical element ofR[d] has the form Σ(r _α d ₁ ^α(1) ...d _n ^α(n) ∣α∈? ⁿ ) withr _αεR. Equality and addition are defined as in commuting polynomial rings, with multiplication induced by the relationsd _i r=rd _i +d _i (r) forrεR and 1≤i≤n. The power series ring $$[[X_1 ,...,X_n ]]R = [[X]]R = \{ \Sigma (X^\alpha r_\alpha \left| {\alpha \in \mathbb{N}^n )} \right|r_\alpha \in R\} $$ is anR[d]-module,d _i acting as partial differentiation ?/?X _i on[[X]]R andR acting via the ring homomorphism $$R \mathrel\backepsilon r \mapsto \Sigma (1/\alpha !) X^\alpha d_1^{\alpha (1)} ...d_n^{\alpha (n)} (r) \in [[X]]R$$ . Then the module[[X]]R is a big injective cogenerator in the sense of Roos [11]. This result is in a certain sense a dual of the Hilbert Basis Theorem: For each left idealL ofR[d] there exists afinite number of power seriesf ₁, ...,f _m such thatL is the annihilator of thef _i inR[d]. For commutative rings of differential operators the minimal injective cogeneratorM is explicitly described as a submodule of[[X]]R, especially forR=? we haveM=Σ(R[X] exp (ΣX _i a _i )|(a ₁, ...,a _n )εR ⁿ ) and all these power series are convergent.     

Similarity invariants for matrices over a commutative Artinian chain ring

Suppose that R is a commutative Artinian chain ring, A is an m × m matrix over R, and S is a discrete valuation ring such that R is a homomorphic image of S. We consider m ideals in the polynomial ring over S that are similarity invariants for matrices over R, i.e., these ideals coincide for similar matrices. It is shown that the new invariants are stronger than the Fitting invariants, and that new invariants solve the similarity problem for 2 × 2 matrices over R.     

Simultaneous orderings in function fields

For every Dedekind domain R, Bhargava defined the factorials of a subset S of R by introducing the notion of p-ordering of S, for every maximal ideal p of R. We study the existence of simultaneous ordering in the case S=R=O_K, where O_K is the ring of integers of a function field K over a finite field F_q. We show, that when O_K is the ring of integers of an imaginary quadratic extension K of F_q(T), K=F_q(T)/(Y²-D(T)), then there exists a simultaneous ordering if and only if degD?1.