scholarly journals Rings with polynomial identity and centrally essential rings

2019 ◽  
Vol 60 (4) ◽  
pp. 657-661 ◽  
Author(s):  
V. T. Markov ◽  
A. A. Tuganbaev
Keyword(s):  
Polynomial Identity

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).

scholarly journals On commutativity of one-sideds-unital rings

1992 ◽  
Vol 15 (4) ◽  
pp. 813-818
Author(s):  
H. A. S. Abujabal ◽  
M. A. Khan
Keyword(s):  
Polynomial Identity ◽  
Unital Ring ◽  
Unital Rings

The following theorem is proved: Letr=r(y)>1,s, andtbe non-negative integers. IfRis a lefts-unital ring satisfies the polynomial identity[xy−xsyrxt,x]=0for everyx,y∈R, thenRis commutative. The commutativity of a rights-unital ring satisfying the polynomial identity[xy−yrxt,x]=0for allx,y∈R, is also proved.

scholarly journals A note on extensions of Baer and P. P. -rings

1974 ◽  
Vol 18 (4) ◽  
pp. 470-473 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Left Ideal ◽  
Polynomial Identity ◽  
Polynomial Rings ◽  
Nilpotent Elements ◽  
Baer Ring ◽  
Baer Rings ◽  
Left Annihilator

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

Retracts and Injectives

1982 ◽  
Vol 25 (4) ◽  
pp. 462-467 ◽  
Author(s):  
Shalom Feigelstock ◽  
Aaron Klein
Keyword(s):  
Polynomial Identity ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Embedding Theorems

AbstractEmbedding theorems are employed to show that many important categories do not possess non-trivial retracts or injectives. E.g., the categories of monoids, groups, rings, rings with unity, polynomial identity rings, nilpotent groups, solvable groups, and several varieties of groups.

scholarly journals Unique factorisation rings

1992 ◽  
Vol 35 (2) ◽  
pp. 255-269 ◽  
Author(s):  
A. W. Chatters ◽  
M. P. Gilchrist ◽  
D. Wilson
Keyword(s):  
Prime Ideal ◽  
Prime Ring ◽  
Polynomial Identity ◽  
Noetherian Ring ◽  
Basic Theory ◽  
Maximal Order ◽  
Prime Element ◽  
Polynomial Extension ◽  
Simple Ring ◽  
Left And Right

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

scholarly journals Left semi-braces and solutions of the Yang–Baxter equation

2019 ◽  
Vol 31 (1) ◽  
pp. 241-263 ◽  
Author(s):  
Eric Jespers ◽  
Arne Van Antwerpen
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Baxter Equation ◽  
Polynomial Algebras ◽  
Finite Set ◽  
Theoretic Solution ◽  
Kirillov Dimension ◽  
Do So

Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

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