We prove that if the compressed zero-divisor graph of a finite associative ring contains only one strong vertex then this vertex has a loop. Moreover, more properties of the compressed …
We prove that if the compressed zero-divisor graph of a finite associative ring contains only one strong vertex then this vertex has a loop. Moreover, more properties of the compressed zero-divisor graph of a finite associative ring are proved.
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices …
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.
Let ra, rb, rc be the radii, and OA, OB, OC the centers of tangent circles at the vertices to the circumcircle of a triangle ABC and to the opposite …
Let ra, rb, rc be the radii, and OA, OB, OC the centers of tangent circles at the vertices to the circumcircle of a triangle ABC and to the opposite sides. In the paper [Andrica D., Marinescu D.S. New interpolation inequalities to Euler's R≥2 // Forum Geometricorum. 2017. Vol. 17], the authors proved that 4/R £ 1/ra + 1/rb +1/rc £2/r. In the paper [Isaev I., Maltsev Yu., Monastyreva A. On some relations in geometry of a triangle // Journal of Classical Geometry. 2018. Vol. 4], it is given the following generalization of these inequalities: 1/ra + 1/rb +1/rc=2/R+1/r. In that paper, we find the area of the triangle OAOBOC (see Theorem 1). We prove some relations for the numbers R-ra, R-rb, R-rc, where R is the circumradius of a triangle ABC. Namely, we find the expressions 1/R-ra+1/R-rb + 1/R-rc и a/R-ra+b/R-rb + c/R-rc by means by the parameters p, R and r (see Theorem 2). We estimate these values (see Theorem 3). Finally, using the results of paper [Maltsev Yu., Monastyreva A. On some relations for a triangle // International Journal of Geometry. 2019. Vol. 8 (1)] and representing the expression of (1-cos(αβ))(1-cos(β-γ))(1-cos(α-γ)) by means of p, R, r, we prove new proof of the fundamental triangle inequality (see Corollary 2).
We describe the zero divisor graph of a commutative finite local rings R of characteristic 2 with Jacobson radical J such that dimF J/J 2 = 2, dimF J 2 …
We describe the zero divisor graph of a commutative finite local rings R of characteristic 2 with Jacobson radical J such that dimF J/J 2 = 2, dimF J 2 /J
To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product …
To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4 Bloomfield , N. , Wickham , C. ( 2010 ). Local rings with genus 2 zero divisor graph . Comm. Alg. 38 ( 8 ): 2965 – 2980 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] these classes include all local rings of order p 4 and p 5 for prime p.
Abstract To each commutative ring R we can associate a graph, the zero divisor graph of R, whose vertices are the zero divisors of R, and such that two vertices …
Abstract To each commutative ring R we can associate a graph, the zero divisor graph of R, whose vertices are the zero divisors of R, and such that two vertices are adjacent if their product is zero. Presently, we enumerate the local finite commutative rings whose zero divisor graphs have orientable genus 2. Key Words: Commutative algebraFinite ringsLocal ringsZero divisor graphs2000 Mathematics Subject Classification: Primary 05C25Secondary 13H9913M99 ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under award number DMS-0552573. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.
The zero-divisor graph Γ(R) of an associative ring R is the graph with all vertices non-zero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are …
The zero-divisor graph Γ(R) of an associative ring R is the graph with all vertices non-zero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge iff xy = 0 or yx = 0 ([10]). In the present paper, we describe all nilpotent finite rings with planar zero-divisor graphs.
We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. …
We study the zero divisor graph determined by equivalence classes of zero divisors of a commutative Noetherian ring R. We demonstrate how to recover information about R from this structure. In particular, we determine how to identify associated primes from the graph.
The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y …
The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all non-zero (one-sided and two-sided) zero-divisors of R, and two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. [S. P. Redmond, The zero-divisor graph of a noncommutative ring, Int. J. Commut. Rings1(4) (2002) 203–211.] In the present paper, all finite rings with Eulerian zero-divisor graphs are described.
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices …
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.
"On Certain Polynomials Associated with the Triangle." Mathematics Magazine, 36(4), pp. 247–248
"On Certain Polynomials Associated with the Triangle." Mathematics Magazine, 36(4), pp. 247–248
The aim of this paper is to classify the zero divisor graph of finite rings of cubefree order. It is proved that all zero divisor graphs can be interpreted as …
The aim of this paper is to classify the zero divisor graph of finite rings of cubefree order. It is proved that all zero divisor graphs can be interpreted as the extended join over well-known graphs.
The zero-divisor graph of an associative ring R is a graph whose vertices are all nonzero (one-sided and two-sided) zero divisors of R , two distinct vertices x, y are …
The zero-divisor graph of an associative ring R is a graph whose vertices are all nonzero (one-sided and two-sided) zero divisors of R , two distinct vertices x, y are connected by an edge if and only if xy = 0 or yx = 0. In this paper, all finite nonnilpotent rings with planar zero-divisor graphs are completely described. In the previous paper by Kuzmina and Maltsev, the finite nilpotent rings with planar zero-divisor graphs were studied. Thus, this paper completes the description of finite rings with planar zero-divisor graphs.
Let R, r, s represent respectively the circumradius, the inradius and the semiperimeter of a triangle with sides a, b, c. Let f(R, r) and F(R, r) be homogeneous real …
Let R, r, s represent respectively the circumradius, the inradius and the semiperimeter of a triangle with sides a, b, c. Let f(R, r) and F(R, r) be homogeneous real functions. Let q(R, r) and Q(R, r) be real quadratic forms. The latter functions are thus a special case of the former. Our main result is to derive the strongest possible inequalities of the form 1 with equality only for the equilateral triangle.
ABSTRACT There is a natural graph associated to the zero-divisors of a commutative ring In this article we essentially classify the cycle-structure of this graph and establish some group-theoretic properties …
ABSTRACT There is a natural graph associated to the zero-divisors of a commutative ring In this article we essentially classify the cycle-structure of this graph and establish some group-theoretic properties of the group of graph-automorphisms We also determine the kernel of the canonical homomorphism from to
It may happen that an element in a ring is both a zero-divisor and an inverse, that it possesses a right-inverse though no left-inverse, and that it is neither a …
It may happen that an element in a ring is both a zero-divisor and an inverse, that it possesses a right-inverse though no left-inverse, and that it is neither a zero-divisor nor an inverse.Thus there arises the problem of rinding conditions assuring the absence of these paradoxical phenomena; and it is the object of the present note to show that chain conditions on the ideals serve this purpose.At the same time we obtain criteria for the existence of unit-elements.The following notations shall be used throughout.The element e in the ring R is a left-unit for the element u in R, if eu = u ; and e is a leftunit for R, if it is a left-unit for every element in R. Right-units are defined in a like manner; and an element is a universal unit f or R, if it is both a right-and a left-unit for R.The element u is a right-zero-divisor, if there exists an element v ^ 0 in R such that vu = 0 ; and u is a right-inverse in R, if there exists an element w in R such that wu is a left-unit for u and a right-unit for R. Left-zero-divisors and left-inverses are defined in a like manner.Note that 0 is a zero-divisor, since we assume that the ring R is different from 0.L(u) denotes the set of all the elements x in R which satisfy xu = 0; clearly L(u) is a left-ideal in the ring R and every left-ideal of the form L(u) shall be termed a zero-dividing left-ideal.Principal leftideals 1 are the ideals of the form Rv for v in R and the ideals vR are the principal right-ideals.
A completely primary finite ring is a ring R with identity 1 ≠ 0 whose subset of all its zero‐divisors forms the unique maximal ideal J . Let R be …
A completely primary finite ring is a ring R with identity 1 ≠ 0 whose subset of all its zero‐divisors forms the unique maximal ideal J . Let R be a commutative completely primary finite ring with the unique maximal ideal J such that J 3 = (0) and J 2 ≠ (0). Then R / J ≅ G F ( p r ) and the characteristic of R is p k , where 1 ≤ k ≤ 3, for some prime p and positive integer r . Let R o = G R ( p k r , p k ) be a Galois subring of R and let the annihilator of J be J 2 so that R = R o ⊕ U ⊕ V , where U and V are finitely generated R o ‐modules. Let nonnegative integers s and t be numbers of elements in the generating sets for U and V , respectively. When s = 2, t = 1, and the characteristic of R is p ; and when t = s ( s + 1)/2, for any fixed s , the structure of the group of units R ∗ of the ring R and its generators are determined; these depend on the structural matrices ( a i j ) and on the parameters p , k , r , and s .
Описание конечных нильпотентных колец, имеющих планарные графы делителей нуля
Описание конечных нильпотентных колец, имеющих планарные графы делителей нуля
In this note, we show a sharpened version of the classical fundamental triangle inequality, as followswhere φ = min
In this note, we show a sharpened version of the classical fundamental triangle inequality, as followswhere φ = min
A geometric approach to the improvement of Blundon's inequalites given in [11] is presented.If φ = min{|A -B|,|B -C|,|C -A|}, then we proved the inequalitycos φ cos ION cos φ …
A geometric approach to the improvement of Blundon's inequalites given in [11] is presented.If φ = min{|A -B|,|B -C|,|C -A|}, then we proved the inequalitycos φ cos ION cos φ , where O is the circumcenter, I is the incenter, and N is the Nagel point of triangle ABC .As a direct consequence, we obtain a sharp version to Gerretsen's inequalities [7].