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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]

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]

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]