Search

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]

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]

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 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]

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]

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 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]

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]

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]

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]

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]

It is well known that if every cyclic right module over a ring is injective, then the ring is semisimple artinian. This classical theorem of Osofsky promoted a considerable interest in the rings whose cyclics satisfy a certain generalized injectivity condition, such as being quasi-injective, continuous, quasi-continuous, or . Here we carry out a study of the rings whose cyclic modules are -modules. The motivation is the observation that a ring is semisimple artinian if and only if every -generated right -module is a -module. Many basic properties are obtained for the rings whose cyclics are -modules, and some structure theorems are proved. For instance, it is proved that a semiperfect ring has all cyclics -modules if and only if it is a direct product of a semisimple artinian ring and finitely many local rings, and that a right self-injective regular ring has all cyclics -modules if and only if it is a direct product of a semisimple artinian ring, a strongly regular ring and a matrix ring over a strongly regular ring. Applications to the rings whose -generated modules are -modules, and the rings whose cyclics are ADS or quasi-continuous are addressed. [ABSTRACT FROM AUTHOR]

This paper discusses a computational treatment of the localization of an affine coordinate ring at a prime ideal and its associated graded algebra with the means of computer algebra. Building on Mora's paper [T. Mora, La queste del Saint : A computational approach to local algebra, Discrete Appl. Math. 33 (1991) 161-190], we present shorter proofs on two of the central statements and expand on the applications touched by Mora: resolutions of ideals, systems of parameters and Hilbert polynomials, as well as dimension and regularity of . All algorithms are implemented in the library graal.lib for the computer algebra system Singular. [ABSTRACT FROM AUTHOR]

Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring. [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]

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 픍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 픍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number. [ABSTRACT FROM AUTHOR]

For an ordinal α, a modular lattice L with 0 and 1 is α-atomic if L has dual Krull dimension α but each interval [0,x] with x < 1 has dual Krull dimension <α. The properties of α-atomic lattices are presented and applied to module theory. The endomorphism ring of certain types of α-atomic modules is a local domain and hence there is a Krull–Schmidt type theorem for those α-atomic modules. [ABSTRACT FROM AUTHOR]