Construction of a quotient ring of ℤ₂ℱ in which a binomial 1+𝓌 is invertible using small cancellation methods

2019 ◽  
pp. 1-76
Author(s):  
A. Atkarskaya ◽  
A. Kanel-Belov ◽  
E. Plotkin ◽  
E. Rips
Keyword(s):  
Quotient Ring ◽  
Small Cancellation

m-potent commutators involving skew derivations and multilinear polynomials

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohammad Ashraf ◽  
Sajad Ahmad Pary ◽  
Mohd Arif Raza
Keyword(s):  
Prime Ring ◽  
Quotient Ring ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Prime Rings ◽  
Skew Derivation ◽  
Martindale Quotient Ring ◽  
Multilinear Polynomials ◽  
The Right ◽  
The Relationship

AbstractLet {\mathscr{R}} be a prime ring, {\mathscr{Q}_{r}} the right Martindale quotient ring of {\mathscr{R}} and {\mathscr{C}} the extended centroid of {\mathscr{R}}. In this paper, we discuss the relationship between the structure of prime rings and the behavior of skew derivations on multilinear polynomials. More precisely, we investigate the m-potent commutators of skew derivations involving multilinear polynomials, i.e.,\big{(}[\delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})]\big{)}^{m}=[% \delta(f(x_{1},\ldots,x_{n})),f(x_{1},\ldots,x_{n})],where {1<m\in\mathbb{Z}^{+}}, {f(x_{1},x_{2},\ldots,x_{n})} is a non-central multilinear polynomial over {\mathscr{C}} and δ is a skew derivation of {\mathscr{R}}.

scholarly journals A note on Noetherian orders in Artinian rings

1979 ◽  
Vol 20 (2) ◽  
pp. 125-128 ◽  
Author(s):  
A. W. Chatters
Keyword(s):  
Noetherian Ring ◽  
Quotient Ring ◽  
Triangular Matrix ◽  
Idempotent Element ◽  
Noetherian Rings ◽  
Central Idempotent ◽  
Matrix Rings ◽  
Regular Elements ◽  
Zero Divisors ◽  
Left And Right

Throughout this note, rings are associative with identity element but are not necessarily commutative. Let R be a left and right Noetherian ring which has an Artinian (classical) quotient ring. It was shown by S. M. Ginn and P. B. Moss [2, Theorem 10] that there is a central idempotent element e of R such that eR is the largest Artinian ideal of R. We shall extend this result, using a different method of proof, to show that the idempotent e is also related to the socle of R/N (where N, throughout, denotes the largest nilpotent ideal of R) and to the intersection of all the principal right (or left) ideals of R generated by regular elements (i.e. by elements which are not zero-divisors). There are many examples of left and right Noetherian rings with Artinian quotient rings, e.g. commutative Noetherian rings in which all the associated primes of zero are minimal together with full or triangular matrix rings over such rings. It was shown by L. W. Small that if R is any left and right Noetherian ring then R has an Artinian quotient ring if and only if the regular elements of R are precisely the elements c of R such that c + N is a regular element of R/N (for further details and examples see [5] and [6]). By the largest Artinian ideal of R we mean the sum of all the Artinian right ideals of R, and it was shown by T. H. Lenagan in [3] that this coincides in any left and right Noetherian ring R with the sum of all the Artinian left ideals of R.

On a conjecture in Artin groups

2021 ◽  
pp. 1-25
Author(s):  
Arye Juhász
Keyword(s):  
Artin Group ◽  
Two Dimensional ◽  
Artin Groups ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Infinite Type ◽  
Trivial Center

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

Powers and conjugacy in small cancellation groups

1975 ◽  
Vol 26 (1) ◽  
pp. 353-360 ◽  
Author(s):  
Leo P. Comerford
Keyword(s):  
Small Cancellation

scholarly journals Projective dimension and regularity of the path ideal of the line graph

2018 ◽  
Vol 17 (04) ◽  
pp. 1850068 ◽  
Author(s):  
Guangjun Zhu
Keyword(s):  
Line Graph ◽  
Quotient Ring ◽  
Projective Dimension ◽  
Algebraic Properties

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen–Macaulay and also provide some exact formulas for the projective dimension and the regularity of these ideals. As some consequences, we give some exact formulas for the depth of these ideals.

scholarly journals A family of two-generator non-Hopfian groups

2017 ◽  
Vol 27 (06) ◽  
pp. 655-675
Author(s):  
Donghi Lee ◽  
Makoto Sakuma
Keyword(s):  
Continued Fraction ◽  
Small Cancellation ◽  
Continued Fraction Expansion ◽  
Link Group

We construct [Formula: see text]-generator non-Hopfian groups [Formula: see text] where each [Formula: see text] has a specific presentation [Formula: see text] which satisfies small cancellation conditions [Formula: see text] and [Formula: see text]. Here, [Formula: see text] is the single relator of the upper presentation of the [Formula: see text]-bridge link group of slope [Formula: see text], where [Formula: see text] and [Formula: see text] in continued fraction expansion for every integer [Formula: see text].

scholarly journals Stanley depth of squarefree Veronese ideals

2013 ◽  
Vol 21 (3) ◽  
pp. 67-72 ◽  
Author(s):  
Mircea Cimpoeas
Keyword(s):  
Quotient Ring ◽  
Stanley Depth

Abstract We compute the Stanley depth for the quotient ring of a square free Veronese ideal and we give some bounds for the Stanley depth of a square free Veronese ideal. In particular, it follows that both satisfy the Stanley's conjecture.

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