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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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

We consider monomial matrices over a commutative local principal ideal ring R of type M(t,k,n) = Φ (Ik 0 0 tIn−k), 0n − 1 and Ik is the identity k×k matrix. In this paper, we indicate a criterion of the hereditary irreducibility of M(t,k,n) in the case t [k⋅(n−k)n +1 ≠ 0 . [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]

In this paper, we study a class of local rings that are not isomorphic to the class of two local rings we have been working on before. This class is also an isomorphism to a special matrix algebra. We construct a projective coordinate space over a module defined on this special algebra class. Specifically, in a 3-dimensional projective coordinate space, the incidence matrix for a line that combines the certain two points and also all points of a line given with the incidence matrix are found by the help of Maple programme. [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]

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]

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]

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]

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]

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]

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 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]

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]

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]

We show that there is no nontrivial group homomorphism over commutative local rings and division rings for , respectively. It gives a negative answer to Ye's problem (see [S. K. Ye, Low-dimensional representations of matrix group actions on CAT(0) spaces and manifolds, J. Algebra 409 (2014) 219-243]) for the above rings. [ABSTRACT FROM AUTHOR]

Let R be a local commutative noetherian ring and HKR the homology ring of the corresponding Koszul complex. We study the homological properties of HKR in particular the Avramov spectral sequence. When the embedding dimension of R is four and when R can be presented with quadratic relations we have found 101 cases where this spectral sequence degenerates and only three cases where it does not degenerate. We also determine completely the Hilbert series of the bigraded Tor of these HKR in Tables A–D at the end of the paper. We also study some higher embedding dimensions. Among the methods used are the programme BERGMAN by Jörgen Backelin et al., the Macaulay2 -package DGAlgebras by Frank Moore, combined with results by Govorov, Clas Löfwall, Victor Ufnarovski and others. [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]

Let $$R$$ be a commutative chain ring. We use a variation of Gröbner bases to study the lattice of ideals of $$R[x]$$ . Let $$I$$ be a proper ideal of $$R[x]$$ . We are interested in the following two questions: When is $$R[x]/I$$ Frobenius? When is $$R[x]/I$$ Frobenius and local? We develop algorithms for answering both questions. When the nilpotency of $$\text {rad}\,R$$ is small, the algorithms provide explicit answers to the questions. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity. The nilpotent graph of R, denoted by ΓN(R), is a graph with vertex set , and two vertices x and y are adjacent if and only if xy is nilpotent, where is nilpotent, for some y ∈ R*}. In this paper, we determine all isomorphism classes of finite commutative rings with identity whose ΓN(R) has genus one. [ABSTRACT FROM AUTHOR]

A ring R is called pseudopolar if for every a ∈ R there exists p2 = p ∈ R such that p ∈ 2(a), a + p ∈ U(R) and akp ∈ J(R) for some positive integer k. Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings, semiregular rings and strongly π-rad clean rings. In this paper, we completely characterize the local rings R for which M2(R) is pseudopolar. [ABSTRACT FROM AUTHOR]

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring. [ABSTRACT FROM AUTHOR]

An element a of a ring R is called quasipolar provided that there exists an idempotent p ∈ R such that p ∈ comm2(a), a + p ∈ U (R) and ap ∈ Bqnil. A ring R is quasipolar in case every element in R is quasipolar. In this paper, we determine conditions under which subrings of 3 x 3 matrix rings over local rings are quasipolar. Namely, if R. is a bleached local ring, then we prove that T3 (R) is quasipolar if and only if R is uniquely bleached. Furthermore, it is shown that Tn(R) is quasipolar if and only if Tn(R[[x]]) is quasipolar for any positive integer [ABSTRACT FROM AUTHOR]

Abstract: We find all irreducible constituents of the Weil representation of a unitary group of rank m associated to a ramified quadratic extension A of a finite, commutative, local and principal ring R of odd characteristic. We show that this Weil representation is multiplicity free with monomial irreducible constituents. We also find the number of these constituents and describe them in terms of Clifford theory with respect to a congruence subgroup. We find all character degrees in the special case when R is a field. [Copyright &y& Elsevier]

A clean ring is one in which every element is a sum of an idempotent and a unit. It is shown that every ring can be embedded in a clean ring as an essential ring extension. It is seen that the centre of a clean ring need not be a clean ring. There is no “clean hull” of a ring. A family of examples is given where there is a ringR, not a clean ring, embedded in a commutative clean ringSso that there is no clean ringT,R ⊆ T ⊆ S, minimal with that property. It is also shown that a commutative pm ring (each prime ideal is contained in a unique maximal ideal) cannot be extended to a clean ring by the adjunction of finitely many central idempotents. [ABSTRACT FROM PUBLISHER]