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This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semiperfect rings that play important roles in noncommutative algebra. First, we define a ring to be semisimple if the left regular module can be decomposed as a finite direct sum of simple submodules and prove that the index set of computable semisimple rings is Σ 2 0 -hard within the index set of computable rings. Second, we define local rings by using equivalent properties of non-left invertible elements of rings and show that the index set of computable local rings is Π 2 0 -hard within the index set of computable rings. Finally, based on the Π 2 0 definition of local rings, computable semiperfect rings can be described by Σ 3 0 formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ 2 0 -hard and Π 2 0 -hard within the index set of computable rings. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity. In this paper, formal expressions of the number of m × n matrices over R of rank r and the number of invertible matrices over R are presented. The number of matrices over R with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity. The prime ideal sum graph of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I, J are adjacent if and only if I + J is a prime ideal of R. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph. [ABSTRACT FROM AUTHOR]

Let F be a field, let D be a local subring of F , and let Val F (D) be the space of valuation rings of F that dominate D. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of F by proving that a subring R of F dominating D is local, residually algebraic over D and integrally closed in F if and only if there is a closed and connected subspace Z of Val F (D) such that R is the intersection of the rings in Z. Consequently, the intersection of the rings in any closed and connected subset of Val F (D) is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian. [ABSTRACT FROM AUTHOR]

This paper studies the computational complexity of subclasses of semiperfect rings from the perspective of computability theory. A ring is semiperfect if the identity can be expressed as a sum of mutually orthogonal local idempotents. Semisimple rings and local rings are typical subclasses of semiperfect rings that play important roles in noncommutative algebra. First, we define a ring to be semisimple if the left regular module can be decomposed as a finite direct sum of simple submodules and prove that the index set of computable semisimple rings is Σ20-hard within the index set of computable rings. Second, we define local rings by using equivalent properties of non-left invertible elements of rings and show that the index set of computable local rings is Π20-hard within the index set of computable rings. Finally, based on the Π20 definition of local rings, computable semiperfect rings can be described by Σ30 formulas. As a corollary, we find that the index set of computable semiperfect rings can be both Σ20-hard and Π20-hard within the index set of computable rings.

In this paper, we extend our investigation about the generalized comaximal graph introduced in Biswas et al. (Discrete Math Algorithms Appl 11(1):1950013, 2019a). The generalized comaximal graph is defined as follows: given a finite commutative ring R, the generalized comaximal graph G(R) is an undirected graph with its vertex set comprising elements of R and two distinct vertices u, v are adjacent if and only if there exists a non-zero idempotent e ∈ R such that u R + v R = e R . In this study, we focus on identifying the rings R for which the graph G(R) exhibits planarity. Moreover, we provide a characterization of the class of ring for which G(R) is toroidal, denoted by γ (G (R)) = 1 . Furthermore, we also evaluate the energy of the graph G(R). Finally, we demonstrate that the graph G(R) is always Hamiltonian for any finite commutative ring R. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by Γ (R) , is the undirected graph whose vertices are the nonzero zero-divisors of R, and the distinct vertices x and y are adjacent if and only if x y = 0 . Here, we classify the local rings with genus three zero-divisor graphs. [ABSTRACT FROM AUTHOR]

An element a ∈ R is strongly rad-clean provided that there exists an idempotent e ∈ R such that a − e ∈ U(R), ae=ea and eae ∈ J(eRe). In this article, we completely determine when a 2 × 2 matrix over a commutative local ring is strongly rad clean. An application to matrices over power-series is also given. [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]

In this paper, we continue to study skew inverse Laurent series ring R((x−1; α,δ)), where R is a ring equipped with an automorphism α and an α-derivation δ. We directly prove that R((x−1; α,δ)) is semiprimitive reduced if and only if R is α-rigid. Also, as an application of our results, by imposing constraints on α and δ, we completely identify the Jacobson radical of R((x−1; α,δ)) whose set of all nilpotent elements has special conditions. Moreover, necessary and sufficient conditions are obtained for the skew inverse Laurent series ring to satisfy a certain ring property which is among being right Artinian, completely primary, right perfect, (semi)local, semiperfect, semiprimary, semiregular, semisimple and strongly regular, respectively. [ABSTRACT FROM AUTHOR]

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible ρ (A) × ρ (A) -submatrix if and only if the determinantal rank ρ (A) and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras. [ABSTRACT FROM AUTHOR]

It is well known that any finite commutative ring is isomorphic to a direct product of local rings via the Chinese remainder theorem. Hence, there is a great significance to the study of character sums over local rings. Character sums over finite rings have applications that are analogous to the applications of character sums over finite fields. In particular, character sums over local rings have many applications in algebraic coding theory. In this paper, we firstly present an explicit description on additive characters and multiplicative characters over a certain local ring. Then we study Gaussian sums, hyper Eisenstein sums and Jacobi sums over a certain local ring and explore their properties. It is worth mentioning that we are the first to define Eisenstein sums and Jacobi sums over this local ring. Moreover, we present a connection between hyper Eisenstein sums over this local ring and Gaussian sums over finite fields, which allows us to give the absolute value of hyper Eisenstein sums over this local ring. As an application, several classes of codebooks with new parameters are presented. [ABSTRACT FROM AUTHOR]

We prove a necessary condition and a sufficient condition for an n × n -matrix A with determinantal rank ρ (A) = t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its t th compound matrix C t (A). In particular, a non-zero n × n -matrix A with ρ (A) = t over a local commutative ring R is strongly regular if and only if Tr (C t (A)) is a unit in R , and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras. [ABSTRACT FROM AUTHOR]

We call a ring R a generalized uniquely clean (or GUC for short) if every not invertible element in R is uniquely clean. Let R be a ring. It is shown that R is GUC if and only if it is a local ring or a uniquely clean ring. Thus the GUC ring is a generalization of the local ring. Some basic properties of GUC rings are proved. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with unity and x = R. We study the probability that the product of two randomly chosen elements (with replacement) of R equals x. We denote this probability by Probx(R). We determine some bounds for this probability and also obtain some characterizations of finite commutative rings based on this probability. Moreover, we determine the explicit computing formulas for Probx(R) when R = Zm × Zn. [ABSTRACT FROM AUTHOR]

The structure of a finite chain ring has already been described by Wirt in 1972 and others later. The purpose of this article is to describe another structure of a finite chain ring as a ring of square matrices over Galois ring using the companion matrix of a certain Eisenstein polynomial over Galois ring. Such a companion matrix generates the unique maximal ideal of the corresponding matrix chain ring.

In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by Γ(R), is the undirected graph whose vertices are the nonzero zero-divisors of R, and the distinct vertices x and y are adjacent if and only if xy=0. Here, we classify the local rings with genus three zero-divisor graphs.

In this paper, we study two types of nonisotropic symplectic graphs over finite commutative rings defined by nonisotropic free submodules of rank 2 and McCoy rank of matrices. We prove that the graphs are quasi-strongly regular or Deza graphs and we find their parameters. The diameter and vertex transitivity are also analyzed. Moreover, we study subconstituents of these nonisotropic symplectic graphs.

An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants p,n,r,k,k′,m up to isomorphism by finite commutative chain rings (k′=1). Thus the classification of chain rings is reduced to that of finite commutative chain rings.

In 1843, Hamilton (1805–1865) discovered the 4-dimensional division algebra H (ℝ) = ℝ ⊕ i ℝ ⊕ j ℝ ⊕ k ℝ over the field ℝ of real numbers. Hamilton's big discovery is the following beautiful multiplications for the basis { 1 , i , j , k } : i 2 = j 2 = k 2 = i j k = − 1. (⋆) H (ℝ) contains the field ℂ of complex numbers; therefore ℝ ⊂ ℂ ⊂ H (ℝ). Frobenius (1849–1917) showed the following outstanding theorem. Theorem (Frobenius). Up to isomorphism, the only finite-dimensional non-commutative division algebra over ℝ is H (ℝ). Starting from given any ring R and a free right R -module R ⊕ i R ⊕ j R ⊕ k R , the quaternion ring H (R) is canonically defined by the multiplications (⋆). For a commutative field F with 2 ≠ 0 , H (F) is a division ring or isomorphic to the ring of 2 × 2 matrices over F. This is a classical theorem. For nonzero a , b in the center of a ring R , the generalized quaternion ring H (R ; a , b) is defined. H (R ; − 1 , − 1) is H (R). In [I. Kikumasa, K. Koike and K. Oshiro, Complex rings and quaternion rings, East-West J. Math. 21 (2019) 1–19; I. Kikumasa, G. Lee and K. Oshiro, Complex Rings, Quaternion Rings and Octonion Rings (Lambert Academic Publishing, 2020); G. Lee and K. Oshiro, Quaternion rings and octonion rings, Front. Math. China 12(1) (2017) 143–155], quaternion rings and generalized quaternion rings over division rings or other rings are studied. Now, in this paper, from ring theoretic viewpoints, we study quaternion rings H (R) and generalized quaternion rings H (R ; a , b) over local rings R. From our results, we can clearly look over several classical results on H (F) and H (F ; a , b) over commutative fields  F. [ABSTRACT FROM AUTHOR]

Let R be a commutative finite local ring. Two enumeration problems over R are presented. We enumerate the matrices over R with a given McCoy rank and a given number of rows of single unit, and the free linear codes over R which have a given rank and a given number of vectors of single unit.

Let R be a commutative local ring whose maximal ideal is generated by a nilpotent element, and Mat (n , R) be the multiplicative monoid of the square matrices of order n over R. In this paper, we provide the construction of Green's ℋ -equivalence classes in the multiplicative monoid Mat (n , R). Then, we enumerate these classes in the special cases R = ℤ / p d ℤ and R = q [ x ] / 〈 x d 〉. [ABSTRACT FROM AUTHOR]

The structure of a finite chain ring has already been described by Wirt in 1972 and others later. The purpose of this article is to describe another structure of a finite chain ring as a ring of square matrices over Galois ring using the companion matrix of a certain Eisenstein polynomial over Galois ring. Such a companion matrix generates the unique maximal ideal of the corresponding matrix chain ring. [ABSTRACT FROM AUTHOR]

We consider monomial matrices over a commutative local principal ideal ring $R$ of type $M(t,k,n)=\Phi\left(\begin{smallmatrix}I_k&0\\0\,\,&tI_{n-k}\end{smallmatrix}\right)$, $0

In this paper, we prove that if R is a local ring of dimension d ≥ 3 , d odd and 1 (d - 1) ! ∈ R then any skew completable unimodular row v ∈ U m d (R [ X ]) is completable. It is also proved that skew completable unimodular rows of size d ≥ 3 over a regular local ring of dimension d are first row of a 2- stably elementary matrix. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. In Biswas et al. (Disc Math Algorithms Appl 11(1):1950013, 2019), we introduced a graph G(R) whose vertices are elements of R and two distinct vertices a, b are adjacent if and only if a R + b R = e R for some nonzero idempotent e in R. Let G ′ (R) be the subgraph of G(R) generated by the non-units of R. In this paper, we characterize those rings R for which the graph G ′ (R) is connected and Eulerian. Also we characterize those rings R for which genus of the graph G ′ (R) is ≤ 2 . Finally, we show that the graph G ′ (R) is a line graph of some graph if and only if R is either a regular ring or a local ring. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring, I an ideal of R and k ≥ 2 a fixed integer. The ideal-based k -zero-divisor hypergraph H I k (R) of R has vertex set Z I (R , k) , the set of all ideal-based k -zero-divisors of R , and for distinct elements x 1 , x 2 , ... , x k in Z I (R , k) , the set { x 1 , x 2 , ... , x k } is an edge in H I k (R) if and only if x 1 x 2 ⋯ x k ∈ I and the product of the elements of any (k − 1) -subset of { x 1 , x 2 , ... , x k } is not in I. In this paper, we show that H I 3 (R) is connected with diameter at most 4 provided that x 2 ∉ I for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of H I k (R) when R is a product of finite fields. Finally, we find some necessary conditions for a finite ring R and a nonzero ideal I of R to have H I 3 (R) planar. [ABSTRACT FROM AUTHOR]

A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples; one of which answers a question of D. Jorgensen in the negative. [ABSTRACT FROM AUTHOR]

We describe the structure of the projective cover of a module M R over a local ring R and its relation with minimal sets of generators of M R . The behaviour of local right perfect rings is completely different from the behaviour of local rings that are not right perfect. [ABSTRACT FROM AUTHOR]

For any prime p and positive integer m, let R be the finite commutative ring F p m + u F p m + v F p m + u v F p m , where u 2 = 0 , v 2 = 0 and u v = v u . Let λ = λ 1 + u λ 2 + v λ 3 + u v λ 4 be a unit of R, where λ 1 , λ 2 , λ 3 , λ 4 ∈ F p m and λ 1 ≠ 0 . We know that λ -constacyclic codes of length p s over R are exactly ideals of the ring R [ x ] ⟨ x p s - λ ⟩ . For all possible values of λ , we study λ -constacyclic codes of length p s over R. We also extend structures of codes from single alphabet to mixed alphabet, and determine separable constacyclic codes of length (p r , p s) over F p m R . [ABSTRACT FROM AUTHOR]

The main aim is to relate the theoretic properties of a commutative ring with properties of graph. For a commutative ring, the set of zero-divisors of denoted by Z(R). A simple graph Г(R) is associated with the vertices which are non zero zero-divisors denoted by Z(R)*=Z(R)-{0}, where for distinct non zero zero-divisors of R x, y, the vertices x and y are connected by an edge if x y=0. This study illustrates the structure of Г(R) and the properties of Z(R). We study when Г(R) can be a complete graph and a star graph and examine the connectivity and diameter and grith of the graph Г(R). We also study Г(R) for non-isomorphic rings. The properties of Г(R), for a commutative ring R and If Z(R) is an annihilator ideal, and for a local ring R with maximal ideal M are given. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. A right R -module M R is said to have Property (A), if each finitely generated ideal I ⊆ Z (M R) has a nonzero annihilator in M R . Evans [Zero divisors in Noetherian-like rings, Trans. Amer. Math. Soc. 155(2) (1971) 505–512.] proved that, over a commutative ring, zero-divisor modules have Property (A). We study and construct various classes of modules with Property (A). Following Anderson and Chun [McCoy modules and related modules over commutative rings, Comm. Algebra 45(6) (2017) 2593–2601.], we introduce G -dual McCoy modules and show that, for every strictly totally ordered monoid G , faithful symmetric modules are G -dual McCoy. We then use this notion to give a characterization for modules with Property (A). For a faithful symmetric right R -module M R and a strictly totally ordered monoid G , it is proved that the right R [ G ] -module M [ G ] R [ G ] is primal if and only if M R is primal with Property (A). [ABSTRACT FROM AUTHOR]

Let F p m be a finite field of cardinality p m and R 3 = F p m [ u ] 〈 u 3 〉 , where p is a prime, m is a positive integer and u 3 = 0. Laaouine et al. (2021) [7] obtained the torsion code and the number of codewords of cyclic codes of length p s over F p m [ u ] 〈 u 3 〉 , but some of the results were wrong, which in this paper, we improve them. By a new method, we compute the number of codewords of these codes. Finally, by using a one-to-one correspondence between cyclic and α -constacyclic codes of length p s over F p m [ u ] 〈 u 3 〉 , we generalize these results to α -constacyclic codes, where α is a non-zero element of the finite field F p m . [ABSTRACT FROM AUTHOR]

Polynomial factoring has famous practical algorithms over fields– finite, rational and p -adic. However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues. For example, x 2 + p mod p 2 is irreducible, but x 2 + p x mod p 2 has exponentially many factors in the input size (which here is logarithmic in p)! We present the first randomized poly(deg ⁡ f , log ⁡ p) time algorithm to factor a given univariate integral polynomial f modulo p k , for a prime p and k ≤ 4. 1 Thus, we solve the open question of factoring modulo p 3 posed in (Sircana, ISSAC'17). Our method reduces the general problem of factoring f mod p k to that of root finding of a related polynomial E (y) mod 〈 p k , φ (x) ℓ 〉 for some irreducible φ mod p. We can efficiently solve the latter for k ≤ 4 , by incrementally transforming E. Moreover, we discover an efficient refinement of Hensel lifting to lift factors of f mod p to those mod p 4 (if possible). This was previously unknown, as the case of repeated factors of f mod p forbids classical Hensel lifting. [ABSTRACT FROM AUTHOR]

By modifying the existing definition of a compressed zero-divisor graph Γ E (K) , we define a compressed zero-divisor graph Θ (K) of a finite commutative unital ring K , where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of Γ E (K)). We prove that this is the best possible compression which induces a functor Θ , and that this functor preserves categorial products (in both directions). We use the structure of Θ (K) to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is describing cyclic codes of length pn over (ℤp)m. We partition ideals of into two general forms and calculate the annihilator of these two forms. In this way, we can introduce an explanation of cyclic codes and the dual code of an arbitrary cyclic code of length pn over (ℤp)m. [ABSTRACT FROM AUTHOR]

It was shown by Galovich that if R is a commutative unique factorization ring (UFR) with identity, then R is a local ring with a nil maximal ideal. In this paper, we generalize Galovich's results to the non-commutative case. [ABSTRACT FROM AUTHOR]

In this paper, we construct directed strongly regular graphs with new parameters by using partial sum families with local rings. 16 families of new directed strongly regular graphs are obtained and the uniform partial sum families are given. Based on the cyclotomic numbers of finite fields, we present two infinite families of directed strongly regular Cayley graphs from semi-direct products of groups. [ABSTRACT FROM AUTHOR]

A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units. In this note, we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property. [ABSTRACT FROM AUTHOR]

Factorizations of ring elements are described by finite chains of principal ideals. We use the description of cyclically presented modules over local rings to study factorization of elements in local rings. [ABSTRACT FROM AUTHOR]

In this paper, we carry the obtained results for a certain class of Moufang-Klingenberg planes co-ordinatized by the local alternative ring of dual numbers, over to a class of Moufang-Klingenberg planes coordinatized by the plural algebra of order m which is a local alternative ring. So, we show that its collineation group is transitive on quadrangles and therefore that coordinatization of the planes is independent of the choice of the coordinatization quadrangle. Moreover, by a conjugacy class we discuss the definition of cross-ratio of four distinct points on the projective lines for the planes. [ABSTRACT FROM AUTHOR]

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]

A collection β = { B 1 , B 2 , ⋯ , B N } of orthonormal bases for ℂL is called approximately mutually unbiased bases if ∣ 〈 u , v 〉 ∣ ≤ 1 L (1 + o (1)) for all u ∈ Bi, v ∈ Bj and 1 ≤ i ≤ j ≤ N. In this paper, the authors construct approximately mutually unbiased bases by using Gauss sums over Frobenius rings. [ABSTRACT FROM AUTHOR]