Notes on the Kernels of Locally Finite Higher Derivations in Polynomial Rings

2016 ◽  
Vol 44 (5) ◽  
pp. 1924-1930
Author(s):  
Hideo Kojima
Keyword(s):  
Polynomial Rings ◽  
Locally Finite

A Property Satisfying Reducedness over Centers

2021 ◽  
Vol 28 (03) ◽  
pp. 453-468
Author(s):  
Hailan Jin ◽  
Tai Keun Kwak ◽  
Yang Lee ◽  
Zhelin Piao
Keyword(s):  
Jacobson Radical ◽  
Regular Ring ◽  
Finite Ring ◽  
Polynomial Rings ◽  
Free Algebras ◽  
Locally Finite ◽  
Nonzero Characteristic

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

scholarly journals Prime ideals in skew Laurent polynomial rings

1993 ◽  
Vol 36 (2) ◽  
pp. 299-317 ◽  
Author(s):  
K. W. Mackenzie
Keyword(s):  
Prime Ideal ◽  
Polynomial Ring ◽  
Commutative Algebra ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Ideal Structure ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Laurent Polynomial Ring

Let R be a commutative ring and {σ1,…,σn} a set of commuting automorphisms of R. Let T = be the skew Laurent polynomial ring in n indeterminates over R and let be the Laurent polynomial ring in n central indeterminates over R. There is an isomorphism φ of right R-modules between T and S given by φ(θj) = xj. We will show that the map φ induces a bijection between the prime ideals of T and the Γ-prime ideals of S, where Γ is a certain set of endomorphisms of the ℤ-module S. We can study the structure of the lattice of Γ-prime ideals of the ring S by using commutative algebra, and this allows us to deduce results about the prime ideal structure of the ring T. As an example, if R is a Cohen-Macaulay ℂ-algebra and the action of the σj on R is locally finite-dimensional, we will show that the ring T is catenary.

Ring properties related to symmetric rings

2014 ◽  
Vol 24 (07) ◽  
pp. 935-967
Author(s):  
Da Woon Jung ◽  
Tai Keun Kwak ◽  
Min Jung Lee ◽  
Yang Lee
Keyword(s):  
Basic Structure ◽  
Ring Theory ◽  
Polynomial Rings ◽  
Minimal Order ◽  
Module Theory ◽  
Finite Rings ◽  
Locally Finite ◽  
Infinite Dimensional ◽  
Finite Dimensional

The study of symmetric rings has important roles in ring theory and module theory. We investigate the structure of ring properties related to symmetric rings and introduce H-symmetric and π-symmetric as generalizations. We construct a non-symmetric reversible ring whose basic structure is infinite-dimensional, comparing with the finite-dimensional such rings of Anderson, Camillo and Marks. The structure of π-reversible rings (with or without identity) of minimal order is completely investigated. The properties of zero-dividing polynomials over IFP rings are studied more to show that polynomial rings over symmetric rings are π-symmetric. It is also proved that all conditions in relation with our arguments in this paper are equivalent for regular or locally finite rings.

Examples of theories of presheaf type

2018 ◽  
Author(s):  
Olivia Caramello
Keyword(s):  
Finite Groups ◽  
Linear Independence ◽  
Geometric Theory ◽  
Vector Spaces ◽  
Linear Orders ◽  
Locally Finite ◽  
Strong Unit ◽  
Locally Finite Groups ◽  
Simplicial Sets ◽  
Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

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