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Let R be a finite ring (with nonzero identity) and let B i (G R) denote the unitary one-matching bi-Cayley graph over R. In this paper, we calculate the chromatic, edge chromatic, clique and independent numbers of B i (G R) and we show that the graph B i (G R) is a strongly regular graph if and only if | R | = 2. Also, we study the perfectness of B i (G R). Moreover, we prove that if B i (G R) ≅ B i (G S) and ω (G R) ≠ 2 , then G R ≅ G S and B i (G R / J R ) ≅ B i (G S / J S ) , where J R and J S are Jacobson radicals of R and S , respectively. Furthermore, for a finite field with ≠ ℤ 2 and a ring S , we prove that if B i (G M n ()) ≅ B i (G S) , where n > 1 is an integer, then S ≅ M n () , and so S is a semisimple ring. [ABSTRACT FROM AUTHOR]

Let R be a finite unitary ring whose group of units is not solvable but all groups of units of all its proper subrings are solvable. In this paper, we classify these rings and show that all finite rings of order pn for n < 5 and some of order p6 are in this class of rings. [ABSTRACT FROM AUTHOR]

Let Cent (R) denote the set of all distinct centralizers in a ring R. A ring R is said to be n -centralizer ring if | Cent (R) | = n , where n ∈ ℕ. In this paper, we illustrate some connections between the centralizers and pairwise non-commuting elements of a finite ring. Besides that, we characterize all n -centralizer finite rings for n = 6 , 7 , 8 , 9 , and also compute the commuting probability for all n -centralizer finite rings for n = 6 , 7 , 8 , 9. [ABSTRACT FROM AUTHOR]

In this paper, we determine the distance Laplacian spectra of graphs obtained by various graph operations. We obtain the distance Laplacian spectrum of the join of two graphs G 1 and G 2 in terms of adjacency spectra of G 1 and G 2 . Then we obtain the distance Laplacian spectrum of the join of two graphs in which one of the graphs is the union of two regular graphs. Finally, we obtain the distance Laplacian spectrum of the generalized join of graphs G i , where 1 ≤ i ≤ n , in terms of their adjacency spectra. As applications of the results obtained, we have determined the distance Laplacian spectra of some well-known classes of graphs, namely the zero divisor graph of ℤ n , the commuting and the non-commuting graph of certain finite groups like D n and Dic n , and the power graph of various finite groups like ℤ n , D n and Dic n . We show that the zero divisor graph and the power graph of ℤ n are distance Laplacian integral for some specific n. Moreover, we show that the commuting and the non-commuting graph of D n and Dic n are distance Laplacian integral for all n ≥ 2. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We calculate the probability when a finite commutative ring is weakly nil-clean in terms of invariants associated only with the given whole ring. This continues the study of our recent paper concerned with the nil-clean case [P. Danchev and M. Samiei, The probability when a finite commutative ring is nil-clean, Trans. A. Razmadze Math. Inst. 176 (2022)]. [ABSTRACT FROM AUTHOR]

In this work, we came to present the elliptic curves over a nonlocal ring A = 2 d [ X ] (X 2 − X) , where 2 d is a finite field of characteristic 2, and d is a prime integer. We consider an elliptic curve given by the Weierstrass equation of the form Y 2 + X Y Z = X 3 + a X 2 Z + b Z 3 , such that a , b are in the ring A , and b is invertible in A , see Tadmori et al. [Normal form of the elliptic curves over the finite ring, J. Math. Syst. Sci. 4 (2014) 194–196]. More precisely, we give a classification of elements of the elliptic curve over this ring, and we construct the group law over it, which allows us to prove same properties of the elliptic group ( E a , b (A) , +) , by showing the isomorphism E a , b (A) ≃ E π 0 (a) , π 0 (b) ( 2 d ) × E π 1 (a) , π 1 (b) ( 2 d ). [ABSTRACT FROM AUTHOR]

Let R be a unitary ring of finite cardinality p β k , where p is a prime number and p ∤ k. We show that if the group of units of R has at most one subgroup of order p , then R ≅ A ⊕ B , where B is a finite ring of order k and A is a ring of cardinality p β which is one of the six explicitly described types. [ABSTRACT FROM AUTHOR]

We characterize the structure of finite unitary rings R in which every proper subring is commutative and show that R is a commutative ring. [ABSTRACT FROM AUTHOR]

Let R be a finite unitary ring such that R = R 0 [ R ∗ ] , where R 0 is the prime ring and R ∗ is not a nilpotent group. We show that if all proper subgroups of R ∗ are nilpotent groups, then the cardinality of R is a power of 2. In addition, if (R / Jac (R)) ∗ is not a p -group, then either R ≅ M 2 (G F (2)) or R ≅ M 2 (G F (2)) ⊕ A , where M 2 (G F (2)) is the ring of 2 × 2 matrices over the finite field G F (2) and A is a direct sum of copies of the finite field G F (2). [ABSTRACT FROM AUTHOR]

We provide a minimal set of generators for the ideal of polynomials in that map the maximal ideal into one of its powers , where is a discrete valuation ring with a finite residue field. We use this to provide a minimal set of generators for the ideal of polynomials in that send to zero, where is a finite commutative local principal ideal ring. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different additive subgroups of a finite non-commutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations. [ABSTRACT FROM AUTHOR]

In this paper we prove that any finite nilpotent alternative ring with planar zero-divisor graph is associative. [ABSTRACT FROM AUTHOR]

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of odd order). We also examine a connection to sets of integer-valued polynomials. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A new method to obtain self-orthogonal codes over finite field F2 is presented. Based on this method, we provide a construction for quantum error-correcting codes starting from cyclic codes over finite ring R = F2 + uF2. As an example, we present infinite families of quantum error-correcting codes which are derived from cyclic codes over the ring R = F2 + uF2. [ABSTRACT FROM AUTHOR]