Rings which are sums of PI subrings

2019 ◽  
Vol 19 (08) ◽  
pp. 2050157
Author(s):  
Marek Kȩpczyk
Keyword(s):  
Polynomial Identity ◽  
Prime Radical ◽  
Additive Subgroup ◽  
Radical Ring

We study rings [Formula: see text] which are sums of a subring [Formula: see text] and an additive subgroup [Formula: see text]. We prove that if [Formula: see text] is a prime radical ring and [Formula: see text] satisfies a polynomial identity, then [Formula: see text] is nilpotent modulo the prime radical of [Formula: see text]. Additionally, we show that if [Formula: see text] is a [Formula: see text] ring, then the prime radical of [Formula: see text] is nilpotent modulo the prime radical of [Formula: see text]. We also obtain a new condition equivalent to Koethe’s conjecture.

Affine algebras of Gelfand-Kirillov dimension one are PI

1985 ◽  
Vol 97 (3) ◽  
pp. 407-414 ◽  
Author(s):  
L. W. Small ◽  
J. T. Stafford ◽  
R. B. Warfield
Keyword(s):  
Polynomial Identity ◽  
Prime Radical ◽  
Finitely Generated ◽  
Finite Module ◽  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension ◽  
Algebra Over A Field

The aim of this paper is to prove:Theorem.Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

ON EXTENSION OF PRIME RADICAL IN 2-PRIMAL NEAR-RINGS

2020 ◽  
Vol 9 (3) ◽  
pp. 1339-1348
Author(s):  
B. Elavarasan ◽  
K. Porselvi and J. Catherine Grace John ◽  
Porselvi J. Catherine Grace John
Keyword(s):  
Prime Radical

Parallel algorithms for matroid intersection and matroid parity

2015 ◽  
Vol 07 (02) ◽  
pp. 1550019
Author(s):  
Jinyu Huang
Keyword(s):  
Parallel Algorithms ◽  
Polynomial Identity ◽  
Black Box ◽  
Matroid Intersection ◽  
Polynomial Identity Testing ◽  
Identity Testing ◽  
Common Base ◽  
Graphic Matroid ◽  
Matroid Parity ◽  
Nc Algorithms

A maximum linear matroid parity set is called a basic matroid parity set, if its size is the rank of the matroid. We show that determining the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity) is in NC2, provided that there are polynomial number of common bases (basic matroid parity sets). For graphic matroids, we show that finding a common base for matroid intersection is in NC2, if the number of common bases is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC algorithms for matroid intersection and matroid parity. We also give a new RNC2 algorithm that finds a common base for graphic matroid intersection. We prove that if there is a black-box NC algorithm for Polynomial Identity Testing (PIT), then there is an NC algorithm to determine the existence of a common base (basic matroid parity set) for linear matroid intersection (linear matroid parity).

TiCp2Cl-Catalyzed Living Radical and Ring Opening Polymerizations Initiated from Epoxides and Aldehydes in the Synthesis of Linear, Graft and Branched Polymers

2004 ◽  
Vol 856 ◽  
Author(s):  
Alexandru D. Asandei ◽  
Isaac W. Moran ◽  
Gobinda Saha ◽  
Yanhui Chen
Keyword(s):  
Electron Transfer ◽  
Ring Opening ◽  
Glycidyl Methacrylate ◽  
Ring Opening Polymerization ◽  
Branched Polymers ◽  
Single Electron Transfer ◽  
Radical Ring ◽  
Polymer Architectures ◽  
Complex Polymer

ABSTRACTTi(III)Cp2Cl-catalyzed radical ring opening (RRO) of epoxides or single electron transfer (SET) reduction of aldehydes generates Ti alkoxides and carbon centered radicals which add to styrene, initiating a radical polymerization. This polymerization is mediate in a living fashion by the reversible termination of growing chains with the TiCp2Cl metalloradical. In addition, polymers or monomers containing pendant epoxide groups (glycidyl methacrylate) can be used as substrates for radical grafting or branching reactions by self condensing vinyl polymerization. In addition, Ti alkoxides generated in situ by both epoxide RRO and aldehyde SET initiate the living ring opening polymerization of ε-caprolactone. Thus, new initiators and catalysts are introduced for the synthesis of complex polymer architectures.

Radical Ring-Opening Copolymerization of 2-Methylene 1,3-Dioxepane and Methyl Methacrylate:  Experiments Originally Designed To Probe the Origin of the Penultimate Unit Effect

1999 ◽  
Vol 32 (5) ◽  
pp. 1332-1340 ◽  
Author(s):  
G. Evan Roberts ◽  
Michelle L. Coote ◽  
Johan P. A. Heuts ◽  
Leesa M. Morris ◽  
Thomas P. Davis
Keyword(s):  
Methyl Methacrylate ◽  
Ring Opening ◽  
Radical Ring

Scope and limitations of the nitroxide-mediated radical ring-opening polymerization of cyclic ketene acetals

2013 ◽  
Vol 4 (17) ◽  
pp. 4776 ◽  
Author(s):  
Antoine Tardy ◽  
Vianney Delplace ◽  
Didier Siri ◽  
Catherine Lefay ◽  
Simon Harrisson ◽  
...  
Keyword(s):  
Ring Opening ◽  
Ring Opening Polymerization ◽  
Radical Ring ◽  
Cyclic Ketene Acetals ◽  
Ketene Acetals

scholarly journals Generalized Higher Derivations on Lie Ideals of Triangular Algebras

2017 ◽  
Vol 25 (1) ◽  
pp. 35-53
Author(s):  
Mohammad Ashraf ◽  
Nazia Parveen ◽  
Bilal Ahmad Wani
Keyword(s):  
Commutative Ring ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Triangular Algebra ◽  
Higher Derivation ◽  
Square Closed Lie Ideal ◽  
Triangular Algebras

Abstract Let be the triangular algebra consisting of unital algebras A and B over a commutative ring R with identity 1 and M be a unital (A; B)-bimodule. An additive subgroup L of A is said to be a Lie ideal of A if [L;A] ⊆ L. A non-central square closed Lie ideal L of A is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on A, every generalized Jordan triple higher derivation of L into A is a generalized higher derivation of L into A.

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