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.

scholarly journals Leavitt path algebras satisfying a polynomial identity

2016 ◽  
Vol 15 (05) ◽  
pp. 1650084 ◽  
Author(s):  
Jason P. Bell ◽  
T. H. Lenagan ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Polynomial Identity ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Graph Theoretic ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Dimension One ◽  
Kirillov Dimension

Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

scholarly journals Prime affine algebras of Gelfand—Kirillov dimension one

1984 ◽  
Vol 91 (2) ◽  
pp. 386-389 ◽  
Author(s):  
L.W Small ◽  
R.B Warfield
Keyword(s):  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
Author(s):  
Xiangui Zhao ◽  
Yang Zhang
Keyword(s):  
Computational Method ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Algebras ◽  
Ore Extensions ◽  
Basis Method ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

scholarly journals NORMAL DOMAINS WITH MONOMIAL PRESENTATIONS

2009 ◽  
Vol 19 (03) ◽  
pp. 287-303 ◽  
Author(s):  
ISABEL GOFFA ◽  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Commutative Algebra ◽  
Class Group ◽  
Semigroup Algebra ◽  
Closed Domain ◽  
Finitely Generated ◽  
Defining Relations ◽  
Integrally Closed ◽  
Algebra Over A Field

Let A be a finitely generated commutative algebra over a field K with a presentation A = K 〈X1,…, Xn | R〉, where R is a set of monomial relations in the generators X1,…, Xn. So A = K[S], the semigroup algebra of the monoid S = 〈X1,…, Xn | R〉. We characterize, purely in terms of the defining relations, when A is an integrally closed domain, provided R contains at most two relations. Also the class group of such algebras A is calculated.

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.

scholarly journals Presentations for Subrings and Subalgebras of Finite CO-Rank

2019 ◽  
Vol 71 (1) ◽  
pp. 53-71
Author(s):  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Lie Algebra ◽  
Noetherian Ring ◽  
Free Lie Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Commutative Noetherian Ring ◽  
Rank 2 ◽  
Finitely Presented ◽  
Algebra Over A Field

Abstract Let $K$ be a commutative Noetherian ring with identity, let $A$ be a $K$-algebra and let $B$ be a subalgebra of $A$ such that $A/B$ is finitely generated as a $K$-module. The main result of the paper is that $A$ is finitely presented (resp. finitely generated) if and only if $B$ is finitely presented (resp. finitely generated). As corollaries, we obtain: a subring of finite index in a finitely presented ring is finitely presented; a subalgebra of finite co-dimension in a finitely presented algebra over a field is finitely presented (already shown by Voden in 2009). We also discuss the role of the Noetherian assumption on $K$ and show that for finite generation it can be replaced by a weaker condition that the module $A/B$ be finitely presented. Finally, we demonstrate that the results do not readily extend to non-associative algebras, by exhibiting an ideal of co-dimension $1$ of the free Lie algebra of rank 2 which is not finitely generated as a Lie algebra.

scholarly journals Standard prime ideals and lying over for finite extensions of Noetherian algebras

1995 ◽  
Vol 37 (3) ◽  
pp. 311-326
Author(s):  
Günter Krause
Keyword(s):  
Prime Factor ◽  
Finite Sequence ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Associated Prime ◽  
Kirillov Dimension

Let k be a field, let R be a noetherian k-algebra of finite Gelfand-Kirillov dimension GK(R), and let M be a finitely generated right R-module. A standard prime factor series for M is a finite sequence of submodules 0 = N0 ⊂ N1 ⊂…⊂ Ni−1 ⊂ Ni ⊂.… ⊂ Nn = M, such that for each i the annihilator Pi = rR (Ni/Ni−1) is the unique associated prime of Ni/Ni−1 and GK(R/Pi)≤ GK(R/Pj) whenever i≤ j. The set of prime ideals arising from such a series is an invariant of M, called the set of standard primes St(M) of M. The concept, inspired by the notion of a standard affiliated series introduced by Lenagan and Warfield in [7], has been developed in [5], where it was shown that St(M) coincides with the set of all those prime ideals that are minimal over the annihilator of a nonzero submodule of M.

Gelfand-Kirillov Dimension is Exact for Noetherian PI Algebras

1984 ◽  
Vol 27 (2) ◽  
pp. 247-250 ◽  
Author(s):  
T. H. Lenagan
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Finitely Generated ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

AbstractIf O → A → C → B → O is a short exact sequence of finitely generated modules over a Noetherian Pi-algebra then we show that GK(C) = max{GK(A), GK(B)}.

On Algebras With Gelfand-Kirillov Dimension One

1993 ◽  
Vol 119 (4) ◽  
pp. 1095
Author(s):  
Shigeru Kobayashi ◽  
Yuji Kobayashi
Keyword(s):  
Dimension One ◽  
Kirillov Dimension
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