scholarly journals On Finite Nilpotent Matrix Groups over Integral Domains

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Dmitry Malinin
Keyword(s):  
General Result ◽  
Nilpotent Groups ◽  
Commutative Rings ◽  
Integral Domains ◽  
Matrix Groups ◽  
Nilpotent Matrix

We consider finite nilpotent groups of matrices over commutative rings. A general result concerning the diagonalization of matrix groups in the terms of simple conditions for matrix entries is proven. We also give some arithmetic applications for representations over Dedekind rings.

scholarly journals Computing in Nilpotent Matrix Groups

2006 ◽  
Vol 9 ◽  
pp. 104-134 ◽  
Author(s):  
A.S. Detinko ◽  
D.L. Flannery
Keyword(s):  
Finite Fields ◽  
Nilpotent Groups ◽  
Independent Interest ◽  
Linear Groups ◽  
Matrix Groups ◽  
Nilpotent Matrix ◽  
Theoretical Results

AbstractWe present algorithms for testing nilpotency of matrix groups over finite fields, and for deciding irreducibility and primitivity of nilpotent matrix groups. The algorithms also construct modules and imprimitivity systems for nilpotent groups. In order to justify our algorithms, we prove several structural results for nilpotent linear groups, and computational and theoretical results for abstract nil-potent groups, which are of independent interest.

Characterizations of Three Classes of Zero-Divisor Graphs

2012 ◽  
Vol 55 (1) ◽  
pp. 127-137 ◽  
Author(s):  
John D. LaGrange
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Central Vertex ◽  
Integral Domains ◽  
Boolean Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

AbstractThe zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

scholarly journals Twisted homological stability for extensions and automorphism groups of free nilpotent groups

2014 ◽  
Vol 14 (1) ◽  
pp. 185-201 ◽  
Author(s):  
Markus Szymik
Keyword(s):  
General Result ◽  
Automorphism Groups ◽  
Free Groups ◽  
Nilpotent Groups ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
Polynomial Coefficients ◽  
Homological Stability

AbstractWe prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups. The proof presented here uses a general result that applies to arbitrary extensions of groups, and that has other applications as well.

Direct Sums of Torsion-Free Covers

1973 ◽  
Vol 25 (5) ◽  
pp. 1002-1005
Author(s):  
Thomas Cheatham
Keyword(s):  
Direct Product ◽  
Commutative Rings ◽  
Integral Domains ◽  
Direct Sums ◽  
Artinian Rings ◽  
Direct Sum ◽  
Special Case ◽  
Torsion Free

In [4, Theorem 4.1, p. 45], Enochs characterizes the integral domains with the property that the direct product of any family of torsion-free covers is a torsion-free cover. In a setting which includes integral domains as a special case, we consider the corresponding question for direct sums. We use the notion of torsion introduced by Goldie [5]. Among commutative rings, we show that the property “any direct sum of torsion-free covers is a torsion-free cover“ characterizes the semi-simple Artinian rings.

scholarly journals The complexity of the equation solvability problem over semipattern groups

2017 ◽  
Vol 27 (02) ◽  
pp. 259-272 ◽  
Author(s):  
Attila Földvári
Keyword(s):  
Original Problem ◽  
Main Idea ◽  
Solvable Groups ◽  
Nilpotent Groups ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Matrix Groups ◽  
Semidirect Products ◽  
Solvability Problem ◽  
Equation Solvability

The complexity of the equation solvability problem is known for nilpotent groups, for not solvable groups and for some semidirect products of Abelian groups. We provide a new polynomial time algorithm for deciding the equation solvability problem over certain semidirect products, where the first factor is not necessarily Abelian. Our main idea is to represent such groups as matrix groups, and reduce the original problem to equation solvability over the underlying field. Further, we apply this new method to give a much more efficient algorithm for equation solvability over nilpotent rings than previously existed.

On the complement of a graph associated with the set of all nonzero annihilating ideals of a commutative ring

2016 ◽  
Vol 08 (03) ◽  
pp. 1650043 ◽  
Author(s):  
S. Visweswaran ◽  
Patat Sarman
Keyword(s):  
Commutative Ring ◽  
Discrete Math ◽  
Commutative Rings ◽  
Simple Graph ◽  
Integral Domains ◽  
Ideal Formula ◽  
Graph Theoretic ◽  
Vertex Set

The rings considered in this paper are commutative with identity which are not integral domains. Recall that an ideal [Formula: see text] of a ring [Formula: see text] is called an annihilating ideal if there exists [Formula: see text] such that [Formula: see text]. As in [M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727–739], for any ring [Formula: see text], we denote by [Formula: see text] the set of all annihilating ideals of [Formula: see text] and by [Formula: see text] the set of all nonzero annihilating ideals of [Formula: see text]. Let [Formula: see text] be a ring. In [S. Visweswaran and H. D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm Appl. 6(4) (2014), Article ID: 1450047, 22pp], we introduced and studied the properties of a graph, denoted by [Formula: see text], which is an undirected simple graph whose vertex set is [Formula: see text] and distinct elements [Formula: see text] are joined by an edge in this graph if and only if [Formula: see text]. The aim of this paper is to study the interplay between the ring theoretic properties of a ring [Formula: see text] and the graph theoretic properties of [Formula: see text], where [Formula: see text] is the complement of [Formula: see text]. In this paper, we first determine when [Formula: see text] is connected and also determine its diameter when it is connected. We next discuss the girth of [Formula: see text] and study regarding the cliques of [Formula: see text]. Moreover, it is shown that [Formula: see text] is complemented if and only if [Formula: see text] is reduced.

scholarly journals Elementary equivalence for finitely generated nilpotent groups and multilinear maps

1998 ◽  
Vol 58 (3) ◽  
pp. 479-493 ◽  
Author(s):  
Francis Oger
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Elementary Equivalence ◽  
Linear Map ◽  
Multilinear Maps

We show that two finitely generated finite-by-nilpotent groups are elementarily equivalent if and only if they satisfy the same sentences with two alternations of quantifiers. For each integer n ≥ 2, we prove the same result for the following classes of structures:(1) the (n + 2)-tuples (A1, …, An+1, f), where A1, …, An+1 are disjoint finitely generated Abelian groups and f: A1 × … × An → An+1 is a n-linear map;(2) the triples (A, B, f), where A, B are disjoint finitely generated Abelian groups and f: An → B is a n-linear map;(3) the pairs (A, f), where A is a finitely generated Abelian group and f: An → A is a n-linear map.In the proof, we use some properties of commutative rings associated to multilinear maps.

scholarly journals Proof of the Combinatorial Nullstellensatz over Integral Domains, in the Spirit of Kouba

10.37236/463 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Peter Heinig
Keyword(s):  
General Result ◽  
Duality Theory ◽  
Integral Domain ◽  
Present Note ◽  
Direct Proof ◽  
Vector Spaces ◽  
Combinatorial Nullstellensatz ◽  
Integral Domains ◽  
Sole Purpose ◽  
Cramer’S Rule

It is shown that by eliminating duality theory of vector spaces from a recent proof of Kouba [A duality based proof of the Combinatorial Nullstellensatz, Electron. J. Combin. 16 (2009), #N9] one obtains a direct proof of the nonvanishing-version of Alon's Combinatorial Nullstellensatz for polynomials over an arbitrary integral domain. The proof relies on Cramer's rule and Vandermonde's determinant to explicitly describe a map used by Kouba in terms of cofactors of a certain matrix. That the Combinatorial Nullstellensatz is true over integral domains is a well-known fact which is already contained in Alon's work and emphasized in recent articles of Michałek and Schauz; the sole purpose of the present note is to point out that not only is it not necessary to invoke duality of vector spaces, but by not doing so one easily obtains a more general result.

scholarly journals On irreducible divisor graphs in commutative rings with zero-divisors

2015 ◽  
Vol 46 (4) ◽  
pp. 365-388
Author(s):  
Christopher Park Mooney
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Integral Domains ◽  
Graph Theoretic ◽  
Zero Divisors ◽  
Classical Paper

In this paper, we continue the program initiated by I. Beck's now classical paper concerning zero-divisor graphs of commutative rings. After the success of much research regarding zero-divisor graphs, many authors have turned their attention to studying divisor graphs of non-zero elements in integral domains. This inspired the so called irreducible divisor graph of an integral domain studied by J. Coykendall and J. Maney. Factorization in rings with zero-divisors is considerably more complicated than integral domains and has been widely studied recently. We find that many of the same techniques can be extended to rings with zero-divisors. In this article, we construct several distinct irreducible divisor graphs of a commutative ring with zero-divisors. This allows us to use graph theoretic properties to help characterize finite factorization properties of commutative rings, and conversely.

scholarly journals On Finite Nilpotent Groups

1960 ◽  
Vol 12 ◽  
pp. 68-72 ◽  
Author(s):  
G. Bachman
Keyword(s):  
Cyclic Group ◽  
General Result ◽  
Sufficient Conditions ◽  
Nilpotent Groups ◽  
Necessary And Sufficient Conditions ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Special Case

It is well known that if (n, ϕ(n)) = 1, where ϕ(n) denotes the Euler ϕ function, then the only group of order n is the cyclic group. This is a special case of a more general result due to Dickson (2, p. 201); namely, ifwhere the pi are distinct primes and each αi > 0, the necessary and sufficient conditions that the only groups of order n are abelian are (1) each αi ≤ 2 and (2) nois divisible by any p1 … , ps.We wish to establish a theorem which includes these two results. We let G(n) equal the number of groups of order n whereand we seek necessary and sufficient conditions on n so thatClearly, this problem is equivalent to finding necessary and sufficient conditions on n so that all existing groups of order n be nilpotent.

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