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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]

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]

A substitution box (S-box) is the only nonlinear component of a symmetric key encryption cipher that directly affects an encryption scheme's performance and security level. So it is appealing to produce S-boxes with good performance and efficiency. Customarily, in algebra, Galois fields are used to generate such structures. It requires the storage memory of size 8 × 2 8 bits. Similarly, in G F (2 9) this figure reaches to 9 × 2 9 bits. In this study, a novel construction of lookup tables over the algebraic structure of non-chain ring R = Z 8 [ u ] < u 3 > = Z 8 + u Z 8 + u 2 Z 8 , (u 3 = 0) of cardinality 512 is presented that holds 9 × 2 8 computer memory calls. Moreover, the use of non-chain ring operations in the building of lookup tables results in a more complex S-box. Afterward, we use this nonlinear layer to produce confusion in a novel image-enciphering scheme. A nonlinear encryption scheme such as a non-chain-ring S-box construction is susceptible to adversaries because of its complex connections between key ciphertexts and cryptographic primitives. With non-chain ring encryption, you can benefit from enhanced security, complexity of algorithm, resistance to modern attacks, as well as improved computing performance. Texture, algebraic, and statistical analyses were performed on the suggested S-box encryption technique. A comparison of the encryption data using the proposed S-box with the literature available S-boxes demonstrates that the suggested S-box is superior and can be employed in various ciphers. The results of the statistical, differential, and security analyses also reveal that our approach is more effective for image encryption. • A novel construction of lookup tables over the algebraic structure of non-chain ring R = Z 8 [ u ] < u 3 > = Z 8 + u Z 8 + u 2 Z 8 , (u 3 = 0) of cardinality 512 is presented that holds 9 × 2 8 bit computer memory. • The use of a non-chain ring operations in the building of lookup tables results in a more complex S-box. • We use this nonlinear layer to produce confusion in a novel image enciphering scheme. • Texture, algebraic, and statistical analyses were performed on the suggested S-box encryption technique. [ABSTRACT FROM AUTHOR]

The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with a and b adjacent if a b = 0. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph , and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the 2-dimensional dot product graph. [ABSTRACT FROM AUTHOR]

We show that a modification of the proof of a result of Gulliksen gives an elementary proof of the following important theorem by Avramov: if f : (A , m , k) → (B , n , l) is a homomorphism of noetherian local rings and B is of finite flat dimension over A , then the homomorphism induced in André-Quillen homology modules H 2 (A , l , l) → H 2 (B , l , l) is injective. [ABSTRACT FROM AUTHOR]

For a ring R (not necessarily commutative) with identity, the comaximal right ideal graph of R , denoted by G ( R ) , is a graph whose vertices are the nonzero proper right ideals of R , and two distinct vertices I and J are adjacent if and only if I + J = R . In this paper we consider a subgraph G * ( R ) of G ( R ) induced by V ( G ( R ) ) ∖ J ( R ) , where J ( R ) is the set of all proper right ideals contained in the Jacobson radical of R . We prove that if R contains a nontrivial central idempotent, then G * ( R ) is a star graph if and only if R is isomorphic to the direct product of two local rings, and one of these two rings has unique maximal right ideal {0}. In addition, we also show that R has at least two maximal right ideals if and only if G * ( R ) is connected and its diameter is at most 3, then completely characterize the diameter of this graph. [ABSTRACT FROM AUTHOR]

We investigate the structure of possibly degenerate ε -hermitian forms over local rings. We prove classification theorems in the cases where the ring is complete and either the form is nondegenerate or the ring is a discrete valuation ring. In the latter case we describe a complete set of invariants for such forms based on a generalisation of the classical notion of the radical of the form. [ABSTRACT FROM AUTHOR]

The study of the class of van der Waerden rings was initiated by Comfort, Remus and Szambien in [7] . A compact ring ( R , T ) is a van der Waerden ring if and only if T is the only totally bounded ring topology on R . In this paper the study of this class of compact rings is continued. We prove that an Abelian compact van der Waerden ring R has the form R = ( ∏ i ∈ I R i ) × L , where L is a compact radical ring with radical topology and R i ( i ∈ I ) is a compact local topologically finitely generated ring. It is shown that every van der Waerden ring is totally disconnected and metrizable. Furthermore, the class of van der Waerden rings is closed under extensions. Different classes of compact rings are studied which are closely related to van der Waerden rings. [ABSTRACT FROM AUTHOR]

It was proved in [4] that every group ring of a torsion abelian group over a commutative local ring is a semi-clean ring. It was asked in [4] whether every group ring of a torsion abelian group over a commutative clean ring is a semi-clean ring and whether every group ring of a torsion abelian group over a commutative semi-clean ring is a semi-clean ring. In this paper, we give a positive answer to question 1 and a negative answer to question 2. [ABSTRACT FROM AUTHOR]

Abstract: Let K s(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K s(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K s(R) to be strongly clean and, respectively, for the ring K s(R) to be strongly clean. Specializing to s =1 yields some known results. New families of strongly clean rings are presented. [Copyright &y& Elsevier]

Abstract: This paper extends Hua’s theorem on the geometry of rectangular matrices over a division ring to the case of Bezout domains. Let be integers , R an be two Bezout domains. Assume that is an adjacency preserving bijective map in both directions. Further, assume that is a local ring, or is an invertibility preserving map, or is an additive map. This paper obtains the algebraic formulas of . As applications, the ring semi-isomorphisms from to are characterized, and the group isomorphisms from to are discussed. [Copyright &y& Elsevier]

Abstract: Let be a commutative, local, and principal ideal ring with maximal ideal and residue class field . Suppose that every element of is square. Then the problem of classifying arbitrary symmetric matrices over by congruence naturally reduces, and is actually equivalent to, the problem of classifying invertible symmetric matrices over by congruence. [Copyright &y& Elsevier]

Abstract: The paper studies the following question: Given a ring , when does the zero-divisor graph have a regular endomorphism monoid? We prove if contains at least one nontrivial idempotent, then has a regular endomorphism monoid if and only if is isomorphic to one of the following rings: ; ; ; , where are fields. In addition, we determine all positive integers for which has the property. [Copyright &y& Elsevier]

Abstract: We choose formal topology to deal in a basic manner with the Zariski spectra of commutative rings and their structure sheaves. By casting prime and maximal ideals in a secondary role, we thus wish to prepare a constructive and predicative framework for abstract algebraic geometry. In contrast to the classical approach, neither points nor stalks need occur, let alone any instance of the axiom of choice. As compared with the topos-theoretic treatments that may be rendered predicative as well, the road we follow is built from more elementary material. The formal counterpart of the structure sheaf which we present first is our guiding example for a notion of a sheaf on a formal topology. We next define the category of formal geometries, a natural abstraction from that of locally ringed spaces. This allows us to eventually phrase and prove, still within the language of opens and sections, the universal property of the Zariski spectrum. Our version appears to be the only one that is explicitly point-free. [Copyright &y& Elsevier]

We present three constructions of partial difference sets (PDS) using different types of finite local rings. The first construction uses homomorphic images of finite Frobenius local rings and generalizes a previous result by the author. The second construction uses finite Frobenius local rings. The third construction uses finite (noncommutative) chain rings and generalizes a recent construction of partial difference sets by Leung and Ma. The three constructions provide many new PDS in nonelementary abelian groups. [Copyright &y& Elsevier]

Let R be a commutative local ring, and M the maximal ideal of R. We prove that if |R/M|>3, then every matrix in SLnR (n&ges;2) is a product of at most [n/2]+2 commutators of transvections and the length [res A/2]+2 cannot be reduced for n=2. [Copyright &y& Elsevier]

While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade 3. We soften the blow by proving that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen–Macaulay local rings of codimension 3. That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification. [ABSTRACT FROM AUTHOR]