Non-trivial Identities Research Articles

English

Let (L,∧, ∨) be a finite lattice with a least element 0. AG(L) is an annihilating-ideal graph of L in which the vertex set is the set of all nontrivial ideals of L, and two distinct vertices I and J are adjacent if and only if I ∧ J = 0. We completely characterize all finite lattices L whose line graph associated to an annihilating-ideal graph, denoted by L(AG(L)), is a planar or projective graph.

A modular supercongruence for _6F_5: An Apéry-like story

We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to $\zeta (3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Ap\'ery numbers and another Ap\'ery-like sequence.

Orbit Codes over M2(ℱq) and Their Homogeneous Distance

Let q be a power of a prime. The lattice of one-sided ideals of the finite unital non-commutative Frobenius ring M2(ℱq) of 2 × 2 matrices over the Galois field (ℱq) is completely analyzed. It turns out that M2(ℱq) is a principal left semi-local ring in which each left ideal is generated by an idempotent element. The explicit forms of the non-trivial idempotents of M2(ℱq) are determined to give q + 1 proper non-trivial left maximal ideals each with q elements. These are exactly the minimal left ideals as well. Using the structure of M2(ℱq) as a partial ordering of ideals, the generalized Möbius and Euler phi functions are applied to derive the explicit form of the homogeneous weight function on M2(ℱq). This weight depends on whether the element is the zero element, a zero divisor or a unit. A zero divisor gives the largest homogeneous weight. Moreover, orbit codes over M2(ℱq) are constructed via the action of the general linear group GL(2, q) on M2(ℱq) by left translation. The orbit determined by a nonzero nonunit idempotent element of M2(ℱq) forms the nonzero elements of a minimal left ideal of M2(ℱq) which are all zero divisors. Consequently, it is shown that the minimum homogeneous distance of the orbit code generated by a nonzero nonunit idempotent element of M2(ℱq) approaches the Plotkin upper bound as the field size q becomes larger. Analogous results are obtained when the lattice of right ideals is considered and the action of GL(2, q) on M2(ℱq) by right translation is used instead.

On Sum Element Ideal Graphs

Let R be a commutative ring with identity and let x be an element of R. The element ideal graph is a graph whose vertex set is the set of non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if xIJ. In this paper, we consider a new kind of graph associated with R denoted by whose vertex set is the set of non-trivial ideals of R and two distinct vertices I and J are adjacent if and only if I+J is a vertex of .

The central vertices and radius of the regular graph of ideals

The regular graph of ideals of the commutative ring $R$‎, ‎denoted by ${Gamma_{reg}}(R)$‎, ‎is a graph whose vertex‎ ‎set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element‎. ‎In this paper‎, ‎it is proved that the radius of $Gamma_{reg}(R)$ equals $3$‎. ‎The central vertices of $Gamma_{reg}(R)$ are determined‎, ‎too‎.

Sturmian words and overexponential codimension growth

Let A be a non necessarily associative algebra over a field of characteristic zero satisfying a non-trivial polynomial identity. If A is a finite dimensional algebra or an associative algebra, it is known that the sequence cn(A), n=1,2,…, of codimensions of A is exponentially bounded.If A is an infinite dimensional non associative algebra such sequence can have overexponential growth. Such phenomenon is present also in the case of Lie or Jordan algebras. In all known examples the smallest overexponential growth of cn(A) is (n!)12.Here we construct a family of algebras whose codimension sequence grows like (n!)α, for any real number α with 0<α<1.

The Embedding of Annihilating-Ideal Graphs Associated to Lattices in the Projective Plane

Let $$(L,\wedge ,\vee )$$ be a lattice with a least element 0. The annihilating-ideal graph of L, denoted by $${\mathbb {AG}}(L)$$ , is a graph whose vertex set is the set of all non-trivial ideals of L and, for every two distinct vertices I and J, I is adjacent to J if and only if $$I\wedge J=\{0\}$$ . In this paper, we completely determine all finite lattices L with projective annihilating-ideal graphs $${\mathbb {AG}}(L)$$ .

Automorphisms of the co-maximal ideal graph over matrix ring

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] be the ring of all [Formula: see text] matrices over [Formula: see text], [Formula: see text] be the set of all nontrivial left ideals of [Formula: see text]. The co-maximal ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph with [Formula: see text] as vertex set and two nontrivial left ideals [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text]. If [Formula: see text], it is easy to see that [Formula: see text] is a complete graph, thus any permutation of vertices of [Formula: see text] is an automorphism of [Formula: see text]. A natural problem is: How about the automorphisms of [Formula: see text] when [Formula: see text]. In this paper, we aim to solve this problem. When [Formula: see text], a mapping [Formula: see text] on [Formula: see text] is proved to be an automorphism of [Formula: see text] if and only if there is an invertible matrix [Formula: see text] and a field automorphism [Formula: see text] of [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text].

Taylor’s Modularity Conjecture and Related Problems for Idempotent Varieties

We provide a partial result on Taylor's modularity conjecture, and several related problems. Namely, we show that the interpretability join of two idempotent varieties that are not congruence modular is not congruence modular either, and we prove an analogue for idempotent varieties with a cube term. Also, similar results are proved for linear varieties and the properties of congruence modularity, having a cube term, congruence $n$-permutability for a fixed $n$, and satisfying a non-trivial congruence identity.

The annihilating graph of a ring

Let A be a commutative ring with unity. The annihilating graph of A, denoted by {{mathbb {G}}}(A), is a graph whose vertices are all non-trivial ideals of A and two distinct vertices I and J are adjacent if and only if {rm Ann}(I){rm Ann}(J)=0. For every commutative ring A, we study the diameter and the girth of {mathbb {G}}(A). Also, we prove that if {mathbb {G}}(A) is a triangle-free graph, then {mathbb {G}}(A) is a bipartite graph. Among other results, we show that if {mathbb {G}}(A) is a tree, then {mathbb {G}}(A) is a star or a double star graph. Moreover, we prove that the annihilating graph of a commutative ring cannot be a cycle. Let n be a positive integer number. We classify all integer numbers n for which {mathbb {G}}({{mathbb {Z}}}_n) is a complete or a planar graph. Finally, we compute the domination number of {mathbb {G}}({mathbb {Z}}_n).

On the genus of the graph associated to a commutative ring

Let [Formula: see text] be a commutative ring with identity. We consider a simple graph associated with [Formula: see text], denoted by [Formula: see text], whose vertex set is the set of all non-trivial ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we characterize the commutative Artinian non-local ring [Formula: see text] for which [Formula: see text] has genus one and crosscap one.

Non-abelian tensor product of residually finite groups

Let $G$ and $H$ be groups that act compatibly on each other. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. Suppose that $G$ is residually finite and the subgroup $[G,H] = \langle g^{-1}g^h \ \mid g \in G, h\in H\rangle$ satisfies some non-trivial identity $f \equiv~1$. We prove that if $p$ is a prime and every tensor has $p$-power order, then the non-abelian tensor product $G \otimes H$ is locally finite. Further, we show that if $n$ is a positive integer and every tensor is left $n$-Engel in $\eta(G,H)$, then the non-abelian tensor product $G \otimes H$ is locally nilpotent. The content of this paper extend some results concerning the non-abelian tensor square $G \otimes G$.

The Mahler measure of a Weierstrass form

We prove an identity between Mahler measures of polynomials that was originally conjectured by Boyd. The combination of this identity with a result of Zudilin leads to a formula involving a Mahler measure of a Weierstrass form of conductor 17 given in terms of [Formula: see text]. Our proof involves a non-trivial identity between regulators which leads to the elliptic curve [Formula: see text]-function being expressed in terms of the regulator evaluated in a non-rational non-torsion point.

Born–Infeld solitons, maximal surfaces, and Ramanujan’s identities

We show that a Born-Infeld soliton can be realised either as a spacelike minimal graph or timelike minimal graph over a timelike plane or a combination of both away from singular points. We also obtain some exact solutions of the Born-Infeld equation from already known solutions to the maximal surface equation. Further we present a method to construct a one-parameter family of complex solitons from a given one parameter family of maximal surfaces. Finally, using Ramanujan's Identities and the Weierstrass-Enneper representation of maximal surfaces, we derive further non-trivial identities.

Reduced Twisted Crossed Products over C*-Simple Groups

We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and maximal invariant ideals of the underlying C*-algebra, and a bijective correspondence between tracial states on the reduced crossed product and invariant tracial states on the underlying C*-algebra. In particular, the reduced crossed product is simple if and only if the underlying C*-algebra has no proper non-trivial invariant ideals, and the reduced crossed product has a unique tracial state if and only if the underlying C*-algebra has a unique invariant tracial state. We also show that the reduced crossed product satisfies an averaging property analogous to Powers' averaging property.

⁎-Superalgebras and exponential growth

In this paper, we study the exponential growth of ⁎-graded identities of a finite dimensional ⁎-superalgebra A over a field F of characteristic zero. If a ⁎-superalgebra A satisfies a non-trivial identity, then the sequence {cngri(A)}n≥1 of ⁎-graded codimensions of A is exponentially bounded and so we study the ⁎-graded exponent expgri(A):=limn→∞⁡cngri(A)n of A. We show that expgri(A)=dimF⁡(A) if and only if A is a simple ⁎-superalgebra and F is the symmetric even center of A. Also, we characterize the finite dimensional ⁎-superalgebras such that expgri(A)≤1 by the exclusion of four ⁎-superalgebras from vargri(A) and construct eleven ⁎-superalgebras Ei,i=1,…,11, with the following property: expgri(A)>2 if and only if Ei∈vargri(A), for some i∈{1,…,11}. As a consequence, we characterize the finite dimensional ⁎-superalgebras A such that expgri(A)=2.

The Element ideal graph Γ(pα qβ ) (Z)

Let R be acommutative ring with identity and let x be an element of R. The Element Ideal Graph Γx (R) is a graph whose vertex set is the set of non-trivial ideals of R and two distinct ideal vertices I and J are adjacent if and only if x∈I J. In this paper we investigate the element ideal graph Γ(pα qβ ) (Z) to explore some of its properties with providing some examples, where Z is the set of integers, α, β∈Z+ and p and q are distinct prime numbers.

On plane polynomial automorphisms commuting with simple derivations

We consider the subgroup Aut(D) consisting of automorphisms of K[x,y] commuting with a derivation D, where K is an algebraically closed field of characteristic 0. We prove that if D is simple (i.e. D does not stabilize non-trivial ideals), then Aut(D)=1. In the case where D is of Shamsuddin type, this result was proven by R. Baltazar in 2014. Moreover, we show that general Shamsuddin type derivations D with Aut(D)=1 are simple.

Co-maximal ideal graphs of matrix algebras

Let R be a ring with unity. The co-maximal ideal graph of R, denoted by \(\Gamma (R)\), is a graph whose vertices are all non-trivial left ideals of R, and two distinct vertices \(I_1\) and \(I_2\) are adjacent if and only if \(I_1 + I_2 = R\). In this paper, some results on the co-maximal ideal graphs of matrix algebras are given. For instance, we determine the domination number, the clique number and a lower bound of the independence number of \(\Gamma (M_n(\mathbb {F}_q))\), where \(M_n(\mathbb {F}_q)\) is the ring of \(n\times n\) matrices over the finite field \(\mathbb {F}_q\). Furthermore, we characterize all rings (not necessarily commutative) whose domination numbers of their co-maximal ideal graphs are finite. Among other results, we show that if \(\Gamma (R)\cong \Gamma (M_n(\mathbb {F}_q))\), where \(n\ge 2\) is a positive integer and R is a ring, then \(R\cong M_n(\mathbb {F}_q)\). Also, it is proved that if R and \(R'\) are two finite reduced rings and \(\Gamma (M_m(R))\cong \Gamma (M_n(R'))\), for some positive integers \(m,n\ge 2\), then \(m=n\) and \(R\cong R'\).

Ramanujan’s identities, minimal surfaces and solitons

Using Ramanujan’s identities and the Weierstrass–Enneper representation of minimal surfaces, and the analogue for Born–Infeld solitons, we derive further non-trivial identities.