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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]

In this paper, we introduce a novel theorem concerning the existence of solutions for Coupled Systems of Time-Fractional Differential Problems. This is achieved through the utilization of the ψ-Caputo fractional derivative, where the order is confined within the range of 0 < α < 1. The validation of the existence result is demonstrated by employing certain conditions, namely Lipschitz and Carathéodory conditions. [ABSTRACT FROM AUTHOR]

Let R be a finite ring. The commuting probability of R is the probability that any two randomly chosen elements of R commute. A ring R is called an n-centralizer ring if it has n distinct centralizers. In this paper, we characterize some n-centralizer finite rings and compute their commuting probabilities. [ABSTRACT FROM AUTHOR]

Let Fq be a finite field, where q is a power of a prime p such that p ≥ 5. Let α be a root of a monic polynomial of a minimal degree over Fq. In this paper, we will study elliptic curves over (Fq[α];+; *), where + is the usual addition and * represent a non-standard product law over Fq[α]. Using elliptic curves over this ring can result in a cryptographic method that is fast, simple and secure. [ABSTRACT FROM AUTHOR]

For any m distinct elements r1, r2, · · ·, rm in a ring R, the m-element centralizer of {r1, r2, · · ·, rm} in R is defined as CR({r1, r2, · · ·, rm}) = {s ∈ R | sr1 = r1s, sr2 = r2s, . . ., srm = rms}, where m ∈ N with m ≽ 2. We define the set of all distinct m-element centralizers in a ring R by m - Cent(R), where m ∈ N with m ≽ 2. Further, R is called (m, n)-centralizer ring if |m - Cent(R)| = n, where n ∈ N. In this paper, we characterize some of the (m, n)-centralizer finite rings for n ≼ 7. [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]

We study some properties of the compressed zero-divisor graph of a finite ring and the partially compressed zero-divisor graph of a finite nilpotent ring. In particular, we describe all nilpotent finite rings whose compressed zero-divisor graphs are some complete graphs with loops. Furthermore, we introduce the notion of partially compressed zero-divisor graph for the nilpotent rings and study the properties of the group. Also, we describe the finite rings whose compressed zero-divisor graph contains a bridge. [ABSTRACT FROM AUTHOR]

Kim and Ohk (2022) showed that binary self-orthogonal [n, k] codes for various dimensions k can be useful for the construction of DNA codes based on quasi self-dual codes over a noncommutative nonunital ring E with four elements. However, there is few classification of binary self-orthogonal codes with dimension ≥ 6 . In this paper, we complete the classification of binary self-orthogonal codes of lengths from 16 to 20 with dimension ≥ 6 . Hence, our classification extends a result of Kim and Ohk (2022). [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]

Let F3d be a finite field of order 3d with d 2 N°. In this paper, we study the elliptic curve over the finite ring F3d ["]:= F3d [X]=(X4-X3), where "4 = "3 of characteristic 3 given by the homogeneous Weierstrass equation of the form Y 2Z = X3 + aX2Z + bZ3; where a; b 2 F3d ["]: Such that we study the arithmetic operations of this ring and define the elliptic curve over it. Next, we show that E-0(a);-0(b)(F3d) and E-1(a);-1(b)(F3d) are two elliptic curves over the finite field F3d, such that 0 is a canonical projection and -1 is a sum projection of coordinate of element in F3d ["] and we conclude by given a classification of elements in elliptic curve over the finite ring F3d ["]. [ABSTRACT FROM AUTHOR]

Von Neumann regularity within a ring of upper triangular matrices is characterized in terms of how such a matrix and an upper triangular generalized inverse can be constructed. In the case of a finite ring, this yields a method for implicitly counting all upper triangular matrices that admit upper triangular generalized inverses. [ABSTRACT FROM AUTHOR]

In this paper, we study the interplay between the structural and spectral properties of the comaximal graph Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) of the ring Zn{{\mathbb{Z}}}_{n} for n>2n\gt 2. We first determine the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) and deduce some of its properties. We then use the structure of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) to deduce the Laplacian eigenvalues of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for various nn. We show that Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) is Laplacian integral for n=pαqβn={p}^{\alpha }{q}^{\beta }, where p,qp,q are primes and α,β\alpha ,\beta are non-negative integers and hence calculate the number of spanning trees of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) for n=pαqβn={p}^{\alpha }{q}^{\beta }. The algebraic and vertex connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) have been shown to be equal for all nn. An upper bound on the second largest Laplacian eigenvalue of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}) has been obtained, and a necessary and sufficient condition for its equality has also been determined. Finally, we discuss the multiplicity of the Laplacian spectral radius and the multiplicity of the algebraic connectivity of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). We then investigate some properties and vertex connectivity of an induced subgraph of Γ(Zn)\Gamma \left({{\mathbb{Z}}}_{n}). Some problems have been discussed at the end of this paper for further research.

We provide a complete answer to the problem of characterizing left Artinian rings which satisfy the (left or right) MacWilliams extension theorem for linear codes, generalizing results of Iovanov [J. Pure Appl. Algebra 220 (2016), pp. 560–576] and Schneider and Zumbrägel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961] and answering the question of Schneider and Zumbragel [Proc. Amer. Math. Soc. 147 (2019), pp. 947–961]. We show that they are quasi-Frobenius rings, and are precisely the rings which are a product of a finite Frobenius ring and an infinite quasi-Frobenius ring with no non-trivial finite modules (quotients). For this, we give a more general "minimal test for injectivity" for a left Artinian ring A: we show that if every injective morphism \Sigma _k\rightarrow A from the k'th socle of A extends to a morphism A\rightarrow A, then the ring is quasi-Frobenius. We also give a general result under which if injective maps N\rightarrow M from submodules N of a module M extend to endomorphisms of M (pseudo-injectivity), then all such morphisms N\rightarrow M extend (quasi-injectivity) and obtain further applications. [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]

Let R be a finite ring and let ${\mathrm {zp}}(R)$ denote the nullity degree of R, that is, the probability that the multiplication of two randomly chosen elements of R is zero. We establish the nullity degree of a semisimple ring and find upper and lower bounds for the nullity degree in the general case. [ABSTRACT FROM AUTHOR]

In this paper, we are going to study the twisted Hessian curves on the local ring F q [ ϵ ] , ϵ 4 = 0 , with F q is a finite field of order q = p b , where p is a prime number ≥ 5 and b ∈ N ∗ . In a first time, we study the arithmetic of the ring F q [ ϵ ] , ϵ 4 = 0 , which will be used in the remainder of this work. After, we define the twisted Hessian curves H a , d 4 over this ring and we give essential properties and the classification of these elements. In addition, we define the group extension H a , d 4 of H a 0 , d 0 by K e r π ~ . We finish this work by introducing a new public key cryptosystem which is a variant of Cramer-Shoup public key cryptosystem on a twisted Hessian curves and study its security and efficiency. Our future work will focus on the generalist these studies for all integers n > 4 , ϵ n = 0 , which is beneficial and interesting in cryptography. [ABSTRACT FROM AUTHOR]

A characterization of idempotency is given for triangular matrices with entries from an arbitrary ring. This yields an elementary approach to constructing such matrices. It also reveals structure allowing for the explicit enumeration or implicit counting of all idempotent triangular matrices over a given finite ring. [ABSTRACT FROM AUTHOR]

In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with n distinct centralizers for any positive integer n = 5. Further we give some connections between the number of distinct centralizers of a finite ring and its commutativity degree. [ABSTRACT FROM AUTHOR]

Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ].

Image encryption based on elliptic curves (ECs) is emerging as a new trend in cryptography because it provides high security with a relatively smaller key size when compared with well-known cryptosystems. Recently, it has been shown that the cryptosystems based on ECs over finite rings may provide better security because they require the computational cost for solving the factorization problem and the discrete logarithm problem. Motivated by this fact, we proposed a novel image encryption scheme based on ECs over finite rings. There are three main steps in our scheme, where, in the first step, we mask the plain image using points of an EC over a finite ring. In step two, we create diffusion in the masked image with a mapping from the EC over the finite ring to the EC over the finite field. To create high confusion in the plain text, we generated a substitution box (S-box) based on the ordered EC, which is then used to permute the pixels of the diffused image to obtain a cipher image. With computational experiments, we showed that the proposed cryptosystem has higher security against linear, differential, and statistical attacks than the existing cryptosystems. Furthermore, the average encryption time for color images is lower than other existing schemes. [ABSTRACT FROM AUTHOR]

We define an indicator of the probability when a finite commutative ring is nil-clean, and calculate this probability for certain classes of finite commutative rings. [ABSTRACT FROM AUTHOR]

Two conditions, (i) and (ii), are defined, that may hold for a given (unital) ring extension R ⊂ S of (unital, associative, not necessarily commutative) finite rings. It is shown that if S is commutative, "either (i) or (ii)" is necessary and sufficient for R ⊂ S to be a minimal ring extension; and for such extensions, (i) and (ii) are logically independent. For extensions with S (finite and) noncommutative, "either (i) or (ii)" is neither necessary nor sufficient for R ⊂ S to be a minimal ring extension; and for such minimal ring extensions, (i) and (ii) are logically independent. Let R ⊂ Sj be minimal ring extensions with Sj (finite and) commutative (for j = 1, 2) and R local. Then: S1 and S2 are the same type (ramified, decomposed or inert) of minimal extension of R ↔ |Z(S1)| = |Z(S2)| ↔ |U(S1)| = |U(S2). [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]

We show that the set of all commuting probabilities in finite rings is a subset of the set of all commuting probabilities in finite nilpotent groups of class 2. These two sets are equal when restricted to groups and rings with odd number of elements. [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]

Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ]. [ABSTRACT FROM AUTHOR]

Let R = F q 2 + u F q 2 + ⋯ + u r - 1 F q 2 be a finite non-chain ring, where q is a prime power, u r = 1 and r | (q + 1) . In this paper, we study u-constacyclic codes over the ring R. Using the matrix of Fourier transform, a Gray map from R to F q 2 r is given. Under the special Gray map, we show that the image of Gray map of u-constacyclic codes over R are cyclic codes over F q 2 , and some new quantum codes are obtained via the Gray map and Hermitian construction from Hermitian dual-containing u-constacyclic codes. [ABSTRACT FROM AUTHOR]

We find bounds for the number of idempotents in a ring of matrices over an arbitrary finite commutative ring. Using these results, we find a bound on the number of idempotents in an arbitrary finite ring, whereby we improve upon the currently known bounds for the number of idempotents. [ABSTRACT FROM AUTHOR]

Several dynamic substitution box (S-box) generators have been proposed in recent years. Some of the existing S-box generators can generate an S-box with good cryptographic properties. However, no analysis has been performed for almost all generators to confirm if they can output highly dynamic S-boxes in a reasonable computational complexity. This article aims to propose an S-box generator that can efficiently generate highly dynamic S-boxes with good cryptographic properties. For this purpose, we use elliptic curves over finite rings. To create randomness in the points on the curves, we define two families of total orders. We then use these ordered curves to generate an S-box of size m using the y-coordinates of an ordered subset of size m of the underlying curve. We performed a rigorous analysis to show that our proposed generator can efficiently generate dynamic S-boxes with good cryptographic properties. Compared to some of the existing generators, it is shown that the proposed S-box generator is more suitable for cryptographic purposes. [ABSTRACT FROM AUTHOR]

In this note we find the least order of a noncommutative even square ring and we note that it is a nil ring having characteristic four. In order to prove the main result given in this note we mainly use suitable examples.

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]

This paper is concerned with, structural properties and construction of quantum codes over Z3 by using constacyclic codes over the finite commutative non-chain ring R = Z3 + vZ3 + ωZ3 + vωZ3 where v2 = 1; ω2 = 1, vω = vω and Z3 is field having 3 elements with characteristic 3. A Gray map is defined between < and Z4 3. The parameters of quantum codes over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3. As an application, some examples of quantum codes of arbitrary length, are also obtained. [ABSTRACT FROM AUTHOR]

Let F3d is the finite field of order 3d with d be a positive integer, we consider A4:= F3d [ε] = 3d [X]=(X4) is a finite quotient ring, where "4 = 0. In this paper, we will show an example of encryption and decryption. Firstly, we will present the elliptic curve over this ring. In addition, we study the algorithmic properties by proposing effective implementations for representing the elements and the group law. Precisely, we give a numerical example of cryptography (encryption and decryption) by using two methods with a secret key. [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 consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables. [ABSTRACT FROM AUTHOR]

We determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring. [ABSTRACT FROM AUTHOR]

Image encryption based on elliptic curves (ECs) is emerging as a new trend in cryptography because it provides high security with a relatively smaller key size when compared with well-known cryptosystems. Recently, it has been shown that the cryptosystems based on ECs over finite rings may provide better security because they require the computational cost for solving the factorization problem and the discrete logarithm problem. Motivated by this fact, we proposed a novel image encryption scheme based on ECs over finite rings. There are three main steps in our scheme, where, in the first step, we mask the plain image using points of an EC over a finite ring. In step two, we create diffusion in the masked image with a mapping from the EC over the finite ring to the EC over the finite field. To create high confusion in the plain text, we generated a substitution box (S-box) based on the ordered EC, which is then used to permute the pixels of the diffused image to obtain a cipher image. With computational experiments, we showed that the proposed cryptosystem has higher security against linear, differential, and statistical attacks than the existing cryptosystems. Furthermore, the average encryption time for color images is lower than other existing schemes.

Let (A, M) be any finite local (nonzero commutative unital) ring. Then the set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a decomposed (minimal ring) extension is in one-to-one correspondence with the collection of sets { I 1 , I 2 } of ideals I 1 and I 2 of A such that M = I 1 ⊕ I 2 (internal direct sum), with any such { I 1 , I 2 } corresponding to the class represented by the ring B = A / I 1 × A / I 2. If A is not a field, then the (finite) set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a ramified (minimal ring) extension has cardinality at least 2; known examples show that this bound cannot be improved. If X denotes an indeterminate over the field F 4 , then R : = F 2 + X F 4 [ X ] / (X 2) is of minimal cardinality in the class of finite local rings A such that A is not a field and the A-algebra isomorphism classes represented by a ramified extension of A constitute fewer than half of the (finitely many) A-algebra isomorphism classes represented by a minimal ring extension of A. [ABSTRACT FROM AUTHOR]

In this paper, we describe such finite non-nilpotent ring that the compressed zero-divisor graph is complete with loops. [ABSTRACT FROM AUTHOR]

Nearly 60 years ago, László Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question of which 2-groups are realizable as unit groups of finite rings, a necessary step toward determining which nilpotent groups are realizable. We prove that all 2-groups of exponent 4 are realizable in characteristic 2. Moreover, while some groups of exponent greater than 4 are realizable as unit groups of rings, we establish general constraints on the exponent of a realizable 2-group. These constraints are used to describe examples of 2-groups that cannot be the group of units of a finite ring. [ABSTRACT FROM AUTHOR]

In this paper, the results introduced in [4] are extended. We define a non trivial automorphism θa over Ra = F4≻ui²= ui,ui uj; ua≼=u2i ujuiff, where i, j = 1, 2, a and a generalized Gray map over Ra which preserves DNA reversibility. The reversibility problem for DNA codes over a family of the finite rings Ra is solved, by using the skew cyclic codes over Ra. 2a-mers are matched with the elements of the finite ring Ra:. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/ and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

We study the compressed zero-divisor graph of a finite associative ring R. In particular, we describe commutative finite rings with compressed zero-divisor graphs of order 2. Moreover, we find all graphs of order 3 that are the compressed zero-divisor graphs of some finite rings. [ABSTRACT FROM AUTHOR]