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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]

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]

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]

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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

This paper contributes several results to the analytic theory of ideals. The main new concept is that of a prereduction (together with the closely related type-prereduction): if R is a ring and A ⊂ I (proper subset) are proper ideals in R, then A is called a prereduction of I in case A is not a reduction of I, but each ideal between A and I is a reduction of I. It is shown that each non-nilpotent proper ideal in a Noetherian ring has at least one prereduction, and a number of the basic properties of the ideals in the set P (I) of all prereductions of I are proved. Also, if I is a non-nilpotent proper ideal in a local (Noetherian) ring (R, M), then: There is a natural one-to-one correspondence between the set E (I) of the equivalence classes of the type-prereductions of I and the set Q (I) of the maximal relevant ideals in the Rees ring of I (that is, the homogeneous prime ideals Q in the ring R [ t − 1 , I t ] (t an indeterminate) such that I t ⊈ Q and the Krull dimension of R [ t − 1 , I t ] / Q is equal to one). P (I n) is the set of maximal elements in { Q [ n ] | Q ∈ Q (I) } for all positive integers n, and, for each Q ∈ Q (I) , Q [ m ] ∈ P (I m) for all large integers m. A complete description is given of the prereductions and the elements in E (I) for each ideal I that is generated by analytically independent elements. If R/M is algebraically closed and I is a non-nilpotent and non-principal ideal in R, then there is a natural one-to-one correspondence between the sets P (I) and E (I) , and P (I n) = { Q [ n ] | Q ∈ Q (I) } for all positive integer n. [ABSTRACT FROM AUTHOR]

The equivalence class[r]o f an element r ∈ R is the set of zero-divisors s such that ann(r) = ann(s), that is, [r] = {s ∈ R : ann(r) = ann(s). The compressed zero-divisor graph, denoted by Γc(R), is the compression of a zero-divisor graph, in which the vertex set is the set of all equivalence classes of nonzero zero-divisors of a ring R, that is, the vertex set of Γc(R) is Re-{[0],[1]}, where Re={[r] :r∈R} and two distinct equivalence classes [r] and [s] are adjacent if and only if rs = 0. In this article, we investigate the planarity of Γc(R) for some finite local rings of order p²,p³ and determine the planarity of compressed zero-divisor graph of some local rings of order 32, whose zero-divisor graph is nonplanar. Further, we determine values of m and n for which Γc(Zn)and Γc(Zn[x]/(xm)) are planar. [ABSTRACT FROM AUTHOR]

In this paper, we introduce n-spaces constructed over an local ring with the maximal ideal (of non-unit elements). So, we give the example of an octonion n-space. Finally, we give two collineations of quaternion n-space. [ABSTRACT FROM AUTHOR]

Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that | B | ≤ N for each ring B such that A ⊂ B is an inert (minimal ring) extension. It follows that A is a (finite) field ⇔ the inert extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ the minimal ring extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ there exists a minimal ring extension A ⊂ B such that B is a flat A-module ⇔ there exists a minimal ring extension A ⊂ B such that B is a local ring and A ⊂ B is a Galois ring extension. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity, n ∈ N and β a bilinear form on R n . In this work, we count the numbers of free submodules and totally isotropic free submodules of R n of rank s by using the lifting idea. We define the graph whose vertex set is the set of totally isotropic free submodules of R n of rank s called the generalized bilinear form graph. We study this graph when (R n , β) is a symplectic space and an orthogonal space. We can determine the degree of each vertex of these graphs. If R is a finite local ring, we show that these graphs are arc transitive and obtain their automorphism groups. We complete our work by proving that we can decompose the graphs over a finite commutative ring into the tensor products of graphs over finite local rings. [ABSTRACT FROM AUTHOR]

An element a in a ring R has a ps-Drazin inverse if there exists b ∈ comm²(a) such that b = bab, (a-ab)k ∈ J(R) for some k ∈ N. Elementary properties of ps-Drazin inverses in a ring are investigated here. We prove that a ∈ R has a ps-Drazin inverse if and only if a has a generalized Drazin inverse and (a-a2)k ∈ J(R) for some k ∈ N. We show Cline's formula and Jacobson's lemma for ps-Drazin inverses. The additive properties of ps-Drazin inverses in a Banach algebra are obtained. Moreover, we completely determine when a 2 × 2 matrix A ∈ M2(R) over a local ring R has a ps-Drazin inverse. [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]

New results are presented and a brief review is given for new methods of the theory of dynamic systems on manifolds over local fields and formal groups over local rings. For the analysis of n-dimensional manifolds and dynamic systems on such manifolds, formal structures are used, in particular, n-dimensional formal groups. Infinitesimal deformations are presented in terms of formal groups. The well-known one-dimensional case is extended and n-dimensional (n ≥ 1) analytic mappings of an open p-adic polydisc (n-disk) D p n are considered. The n-dimensional analogs of modules arising in formal and non-Archimedean dynamic systems are introduced and investigated and their formal-algebraic structure is presented. Rigid structures, objects, and methods are outlined. From the point of view of systems analysis, new, namely formal and non-Archimedean, faces and structures of systems, mappings and iterations of mappings between these faces and structures are introduced and investigated. [ABSTRACT FROM AUTHOR]

This paper presents a new and comparatively secure watermarking technique, in the frequency domain. Our scheme deploys a local ring-based substitution box (S-box). The algebraic algorithm used to synthesize S-box basically exploits one–one correspondence between the multiplicative group of units of the local ring Z 512 and the Galois field F 256 . This S-box has high confusion creating capability due to the structural properties of the local ring and fulfills the necessary requirements to be reliably used in multimedia applications. We use this S-box in a watermarking scheme to make our technique more confusing and secure to provide more support in copyrights protection strategies. The proposed non-blind digital watermarking technique deals with the application of discrete cosine transform (DCT) in the frequency domain which is comparatively more robust than spatial domain techniques. In the proposed scheme, first the watermark image is substituted through the S-box, and the scrambled watermark is then embedded in the DCT-transformed host image.' To measure the strength of the proposed technique, simulation results and statistical analyses are made. Most significant analyses techniques including measures of homogeneity, contrast, energy, entropy, correlation, mean squared error and peak signal to noise ratio are applied which show coherent results. To determine the robustness of our is effectively strong. [ABSTRACT FROM AUTHOR]

Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition problem of the special linear Lie algebra over a finite commutative ring with identity. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1. Let G (R) be a graph with vertices as elements of R , where two distinct vertices a and b are adjacent if and only if a R + b R = e R for some non-zero idempotent e in R. In this paper, we establish a relation between completeness of the graph G (R) and regularity of the ring R. For a finite commutative ring R with 1 , we show that the chromatic number of G (R) is equal to the number of regular elements in R. Moreover, we characterize some graph theoretic properties of G (R) and finally we characterize Eulerian property of the graph G (R). [ABSTRACT FROM AUTHOR]

We classify all non-degenerate skew-hermitian forms defined over certain local rings, not necessarily commutative, and study some of the fundamental properties of the associated unitary groups, including their orders when the ring in question is finite. [ABSTRACT FROM AUTHOR]

We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h−1(𝔭) where 𝔭 2 Spec B. [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]

An element a of an associative ring R is said to be quasinilpotent if 1-ax is invertible for every x ∊ R with xa = ax. Nilpotents and elements in the Jacobson radical of a ring are well-known examples of quasinilpotents. In this paper, properties and examples of quasinilpotents in a ring are provided, and the set of quasinilpotents is applied to characterize rings with some certain properties. [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]

We study the Poincaré series of modules over a fiber product of commutative local rings. We introduce the notion of a weak complete intersection ideal; these are the ideals with the property that every differential in their minimal free resolutions can be represented by a matrix whose entries are in the ideal itself. We show that many properties of the Poincaré series that are known to hold for a fiber product over the maximal ideal also hold for those over weak intersection ideals. [ABSTRACT FROM AUTHOR]

We prove that if R is a commutative, reduced, local ring, then R is Hopfian if and only if the ring R[ x] is Hopfian. This answers a question of Varadarajan [16], in the case when R is a reduced local ring. We provide examples of non-Noetherian Hopfian commutative domains by proving that the finite dimensional domains are Hopfian. Also, we derive some general results related to Hopfian rings. [ABSTRACT FROM AUTHOR]

It is known that a ring is semiprime right Goldie if and only if is a strongly compressible module if and only if every right ideal of is strongly compressible, and that a ring is semiprime Goldie ring if and only if every right ideal of is weakly injective if and only if every essential right ideal of is weakly injective. In this paper, we characterize the rings whose proper (principal) right ideals are strongly compressible, and the rings whose proper (essential) right ideals are weakly injective. [ABSTRACT FROM AUTHOR]

Let $R$ be a finite commutative ring of odd characteristic and let $V$ be a free $R$-module of finite rank. We classify symmetric inner products defined on $V$ up to congruence and find the number of such symmetric inner products. Additionally, if $R$ is a finite local ring, the number of congruent symmetric inner products defined on $V$ in each congruence class is determined. [ABSTRACT FROM AUTHOR]

An element a ∈ R is very clean provided that there exists an idempotent e ∈ R such that ae = ea and either a - e or a + e is invertible in R. A ring R is very clean in case every element in R is very clean. We explore the necessary and sucient conditions under which a triangular 2 x 2 matrix ring over local rings is very clean. The very clean 2 x 2 matrices over commutative local rings are completely determined. Applications to matrices over power series are also obtained. [ABSTRACT FROM AUTHOR]

This is a study of rings whose cyclic modules are -modules with applications to rings whose cyclics are quasi-discrete and, respectively, discrete. [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]

In this paper we study rings R with the property that every finitely generated ideal of R consisting entirely of zero divisors has a nonzero annihilator. The class of commutative rings with this property is quite large; for example, noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose classical ring of quotients are von Neumann regular. We continue to study conditions under which right mininjective rings, right FP-injective rings, right weakly continuous rings, right extending rings, one sided duo rings, semiregular rings and semiperfect rings have this property. [ABSTRACT FROM AUTHOR]

Letkbe a field with a real valuationνandRak-algebra. We show that there exist ak-algebraKand a valuationμonKextendingνsuch that any real valuation ofRis induced byμthrough some homomorphism fromRtoK. Letνbe trivial andRa complete local Noetherian ring with the residue fieldk. LetKbe the ringof Hahn series with its natural valuationμand coefficients in. We prove the following weak universality property: For any local valuationvand a finite set of elementsofR, there exists a homomorphismf:R→Ksuch that,i = 1,…,n. This implies that iffor an idealI, then every point of the local tropicalization ofIlifts to aK-point ofR. [ABSTRACT FROM PUBLISHER]