scholarly journals Lattice-ordered power series fields

1992 ◽  
Vol 52 (3) ◽  
pp. 299-321 ◽  
Author(s):  
R. H. Redfield
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Algebraic Closure ◽  
Torsion Subgroup ◽  
Representation Theorems ◽  
Ordered Field ◽  
Zero Divisors ◽  
Ordered Fields ◽  
Factor Set ◽  
Group Modulo

AbstractA lattice-ordered power series algebra of a totally ordered field over a rooted abelian group may be constructed in a way that is arbitrary only in requiring that a factor set be chosen in the field and an extended total order be chosen on the group modulo its torsion subgroup. The resulting algebra is a field if and only if the subalgebra of elements with torsion support form a field. It follows that if the torsion subgroup may be independently embedded in the algebraic closure of the totally ordered field, or if the resulting algebra has no zero-divisors, then the algebra is a field. The set of supporting subsets for the power series may be characterized abstractly in such a way that previous representation theorems of lattice-ordered fields into power series algebras may be applied to produce representations into power series fields.

On Ore extension and skew power series rings with some restrictions on zero-divisors

2016 ◽  
Vol 16 (09) ◽  
pp. 1750164
Author(s):  
E. Hashemi ◽  
A. As. Estaji ◽  
A. Alhevaz
Keyword(s):  
Power Series ◽  
Left Zero ◽  
Ore Extension ◽  
Series Ring ◽  
Zero Divisors ◽  
Open Questions ◽  
Ore Extensions ◽  
Power Series Rings ◽  
Extension Ring ◽  
The Right

The study of rings with right Property ([Formula: see text]), has done an important role in noncommutative ring theory. Following literature, a ring [Formula: see text] has right Property ([Formula: see text]) if every finitely generated two-sided ideal consisting entirely of left zero-divisors has a nonzero right annihilator. Our results in this paper concerns the right Property ([Formula: see text]) of Ore extensions as well as skew power series rings. We will show that if [Formula: see text] is a right duo ring, then the skew power series ring [Formula: see text] has right Property ([Formula: see text]), when [Formula: see text] is right Noetherian and [Formula: see text]-compatible. Moreover, for a right duo ring [Formula: see text] which is [Formula: see text]-compatible, it is shown that (i) the Ore extension ring [Formula: see text] has right Property ([Formula: see text]) and (ii) [Formula: see text] is right zip if and only if [Formula: see text] is right zip. As a corollary of our results, we provide answers to some open questions related to Property [Formula: see text], raised in [C. Y. Hong, N. K. Kim, Y. Lee and S. J. Ryu, Rings with Property ([Formula: see text]) and their extensions, J. Algebra 315 (2007) 612–628].

Finite Spaces of Signatures

1989 ◽  
Vol 41 (5) ◽  
pp. 808-829 ◽  
Author(s):  
Victoria Powers
Keyword(s):  
Quadratic Forms ◽  
Local Rings ◽  
Character Group ◽  
Witt Rings ◽  
Ordered Field ◽  
Spaces Of Orderings ◽  
Ordered Fields ◽  
Skew Fields ◽  
Finite Spaces ◽  
Field Setting

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

On the Square Subgroup of a Mixed SI-Group

2018 ◽  
Vol 61 (1) ◽  
pp. 295-304 ◽  
Author(s):  
R. R. Andruszkiewicz ◽  
M. Woronowicz
Keyword(s):  
Abelian Group ◽  
Quotient Group ◽  
Abelian Groups ◽  
Additive Group ◽  
Negative Solution ◽  
New Class ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free ◽  
Group Modulo

AbstractThe relation between the structure of a ring and the structure of its additive group is studied in the context of some recent results in additive groups of mixed rings. Namely, the notion of the square subgroup of an abelian group, which is a generalization of the concept of nil-group, is considered mainly for mixed non-splitting abelian groups which are the additive groups only of rings whose all subrings are ideals. A non-trivial construction of such a group of finite torsion-free rank no less than two, for which the quotient group modulo the square subgroup is not a nil-group, is given. In particular, a new class of abelian group for which an old problem posed by Stratton and Webb has a negative solution, is indicated. A new, far from obvious, application of rings in which the relation of being an ideal is transitive, is obtained.

scholarly journals When are dense subgroups of LCA groups closed under intersections?

1994 ◽  
Vol 49 (1) ◽  
pp. 59-67
Author(s):  
M.A. Khan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Compact Abelian Group ◽  
Additive Group ◽  
Locally Compact Abelian Group ◽  
Torsion Subgroup ◽  
Dense Subgroup ◽  
Locally Compact ◽  
Dense Subgroups ◽  
Lca Groups

Let G be a nondiscrete locally compact Hausdorff abelian group. It is shown that if G contains an open torsion subgroup, then every proper dense subgroup of G is contained in a maximal subgroup; while if G has no open torsion subgroup, then it has a dense subgroup D such that G/D is algebraically isomorphic to R, the additive group of reals. With each G, containing an open torsion subgroup, we associate the least positive integer n such that the nth multiple of every discontinuous character of G is continuous. The following are proved equivalent for a nondiscrete locally compact abelian group G:(1) The intersection of any two dense subgroups of G is dense in G.(2) The intersection of all dense subgroups of G is dense in G.(3) G contains an open torsion subgroup, and for each prime p dividing the positive integer associated with G, pG is either open or a proper dense subgroup of G.Finally, we construct a locally compact abelian group G with infinitely many dense subgroups satisfying the three equivalent conditions stated above.

On the notion of algebraic closedness for noncommutative groups and fields

1971 ◽  
Vol 36 (3) ◽  
pp. 441-444 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Wide Class ◽  
First Order ◽  
Division Rings ◽  
Ordered Field ◽  
Ordered Fields ◽  
Skew Fields

The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups.

Zero-divisors and zero-divisor graphs of power series rings

2015 ◽  
Vol 65 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Amor Haouaoui ◽  
Ali Benhissi
Keyword(s):  
Power Series ◽  
Zero Divisor ◽  
Zero Divisors ◽  
Power Series Rings

Finitely continuous differentials on generalized power series

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
I-Chiau Huang
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Vector Space ◽  
Inversion Formula ◽  
Finitely Generated ◽  
Generalized Power Series

AbstractLet κ[[eG]] be the field of generalized power series with exponents in a totally ordered Abelian group G and coefficients in a field κ. Given a subgroup H of G such that G/H is finitely generated, we construct a vector space ΩG/H of differentials as a universal object in certain category of κ[[eH]]-derivations on κ[[eG]]. The vector space ΩG/H together with logarithmic residues gives rise to a framework for certain combinatorial phenomena, including the inversion formula for diagonal delta sets.

scholarly journals On the content of polynomials over semirings and its applications

2016 ◽  
Vol 15 (05) ◽  
pp. 1650088 ◽  
Author(s):  
Peyman Nasehpour
Keyword(s):  
Power Series ◽  
Prime Ideal ◽  
Formal Power Series ◽  
Finite Union ◽  
Distributive Lattices ◽  
Prime Ideals ◽  
Zero Divisors ◽  
Formal Power

In this paper, we prove that Dedekind–Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property ([Formula: see text]) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.

VALUATIONS OF k((X1,…,Xn)) WITH PREASSIGNED GROUP OF VALUES

2004 ◽  
Vol 03 (04) ◽  
pp. 453-468 ◽  
Author(s):  
A. GRANJA
Keyword(s):  
Abelian Group ◽  
Power Series ◽  
Algebraic Extension ◽  
Power Series Ring ◽  
Quotient Field ◽  
Series Ring ◽  
Rational Rank

Let Γ be a totally ordered abelian group of finite rational rank r. Consider s and n two non-negative integers with r+s≤n and denote by K a field. The main purpose of this paper is to construct a s-dimensional valuation v of the quotient field K((X1,…,Xn)) of the corresponding power series ring K[[X1,…,Xn]] such that v birrationally dominates K[[X1,…,Xn]] and has Γ as group of values. This is possible except for the case in which Γ=ℤ is the group of the integers, s=0, n≥2 and K does not admit an infinite algebraic extension. In this last case, we show that there does not exist such a valuation.

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