Semilocal Ring Research Articles

On quadratic forms over semilocal rings

Using a recent result of Panin and Pimenov we show that several results, as for instance the linkage principle, in the algebraic theory of quadratic forms over fields also hold for quadratic forms over regular semilocal domains which contain a field of characteristic not 2. As an application we prove that the Arason and Elman presentation of the powers of the fundamental ideal of the Witt ring of a field extends to semilocal rings which contain an infinite field of characteristic not 2.

The M-intersection graph of ideals of a commutative ring

Let [Formula: see text] be a commutative ring and [Formula: see text] be an [Formula: see text]-module, and let [Formula: see text] be the set of all nontrivial ideals of [Formula: see text]. The [Formula: see text]-intersection graph of ideals of [Formula: see text], denoted by [Formula: see text], is a graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. For every multiplication [Formula: see text]-module [Formula: see text], the diameter and the girth of [Formula: see text] are determined. Among other results, we prove that if [Formula: see text] is a faithful [Formula: see text]-module and the clique number of [Formula: see text] is finite, then [Formula: see text] is a semilocal ring. We denote the [Formula: see text]-intersection graph of ideals of the ring [Formula: see text] by [Formula: see text], where [Formula: see text] are integers and [Formula: see text] is a [Formula: see text]-module. We determine the values of [Formula: see text] and [Formula: see text] for which [Formula: see text] is perfect. Furthermore, we derive a sufficient condition for [Formula: see text] to be weakly perfect.

Additive Maps on Units of Rings

AbstractLet R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

Orbit Codes over M2(ℱq) and Their Homogeneous Distance

Let q be a power of a prime. The lattice of one-sided ideals of the finite unital non-commutative Frobenius ring M2(ℱq) of 2 × 2 matrices over the Galois field (ℱq) is completely analyzed. It turns out that M2(ℱq) is a principal left semi-local ring in which each left ideal is generated by an idempotent element. The explicit forms of the non-trivial idempotents of M2(ℱq) are determined to give q + 1 proper non-trivial left maximal ideals each with q elements. These are exactly the minimal left ideals as well. Using the structure of M2(ℱq) as a partial ordering of ideals, the generalized Möbius and Euler phi functions are applied to derive the explicit form of the homogeneous weight function on M2(ℱq). This weight depends on whether the element is the zero element, a zero divisor or a unit. A zero divisor gives the largest homogeneous weight. Moreover, orbit codes over M2(ℱq) are constructed via the action of the general linear group GL(2, q) on M2(ℱq) by left translation. The orbit determined by a nonzero nonunit idempotent element of M2(ℱq) forms the nonzero elements of a minimal left ideal of M2(ℱq) which are all zero divisors. Consequently, it is shown that the minimum homogeneous distance of the orbit code generated by a nonzero nonunit idempotent element of M2(ℱq) approaches the Plotkin upper bound as the field size q becomes larger. Analogous results are obtained when the lattice of right ideals is considered and the action of GL(2, q) on M2(ℱq) by right translation is used instead.

Generalizations of Rad-supplemented modules

Let R be an associative ring with identity. We introduce the notion of semi-τ-supplemented modules, which is adapted from srs-modules, for a preradical τ on R-Mod. We provide basic properties of these modules. In particular, we study the objects of R-Mod for τ = Rad. We show that the class of semi-τ-supplemented modules is closed under finite sums and factor modules. We prove that, for an idempotent preradical τ on R-Mod, a module M is semi-τ-supplemented if and only if it is τ-supplemented. For τ = Rad, over a local ring every left module is semi-Rad-supplemented. We also prove that a commutative semilocal ring whose semi-Rad-supplemented modules are a direct sum of w-local left modules is an artinian principal ideal ring.

The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

Let $R$ be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over $R$ remain anisotropic after base change to any odd-degree finite étale extension of $R$. This generalization of the classical Artin-Springer theorem (concerning the situation where $R$ is a field) was previously established in the case where all residue fields of $R$ are infinite by I. Panin and U. Rehmann. The more general result presented here permits one to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic $\neq 2$ to the case where the ring has at least one residue field which is finite.

On right G-semilocal rings

ABSTRACTThe notion of semilocal ring is extended to the classes of right G-semilocal and right N-semilocal rings. We explore the algebraic properties of such classes and study their relations with many other rings such as clean, exchange and I-finite rings. Localization of right G-semilocal rings is considered.

Eulerian graphs and automorphisms of a maximal graph

Let R be a commutative ring with identity. Let Γ(R) denote the maximal graph corresponding to the non-unit elements of R, i.e., Γ(R) is a graph with vertices the non-unit elements of R, where two distinct vertices a and b are adjacent if and only if there is a maximal ideal of R containing both. In this paper, we have shown that, for any finite ring R which is not a field, Γ(R) is a Euler graph if and only if R has odd cardinality. Moreover, for any finite ring R ≅ R 1×R 2× · · · ×R n, where the R i is a local ring of cardinality p i αi for all i, and the p i’s are distinct primes, it is shown that Aut(Γ(R)) is isomorphic to a finite direct product of symmetric groups. We have also proved that clique(G(R)’) = χ(G(R)’) for any semi-local ring R, where G(R)’ denote the comaximal graph associated to R.

NORMALIZING EXTENSIONS OF RIGHT G-SEMILOCAL AND RIGHT N-SEMILOCAL RINGS

The aim of this paper is to focus on two new classes of rings called right G-semilocal and right N-semilocal. These classes are a generalization of the class of semilocal rings. We also study the transfer of right G-semilocality and right N-semilocality from a ring R to its normalizing extension S and vice versa.

Cyclic codes and [formula omitted]-constacyclic codes over [formula omitted

Let R=Fp+uFp+vFp+uvFp, where p is a prime and u2=u,v2=v,uv=vu. It is a semi-local ring, not a chain ring. In this paper, some properties of linear codes over the ring R are studied. The generator of cyclic codes and λ1+λ2u+λ3v+λ4uv-constacyclic codes over R are given. A necessary and sufficient condition for self-dual cyclic codes over R is given.

Annihilator-stability and unique generation

A ring R is said to be left uniquely generated if Ra=Rb in R implies that a=ub for some unit u in R. These rings have been of interest since Kaplansky introduced them in 1949 in his classic study of elementary divisors. Writing l(b)={r∈R|rb=0}, a theorem of Canfell asserts that R is left uniquely generated if and only if, whenever Ra+l(b)=R where a,b∈R, then a−u∈l(b) for some unit u in R. By analogy with the stable range 1 condition we call a ring with this property left annihilator-stable. In this paper we exploit this perspective on the left UG rings to construct new examples and derive new results. For example, writing J for the Jacobson radical, we show that a semiregular ring R is left annihilator-stable if and only if R/J is unit-regular, an analogue of Bass' theorem that semilocal rings have stable range 1.

A Bloch–Wigner exact sequence over local rings

In this article we extend the Bloch–Wigner exact sequence over local rings, where their residue fields have more than nine elements. Moreover, we prove Van der Kallen's theorem on the presentation of the second K-group of local rings such that their residue fields have more than four elements. Note that Van der Kallen proved this result when the residue fields have more than five elements. Although we prove our results over local rings, all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings.

On J(R) of the semilocal rings

On J(R) of the semilocal rings

Module Decompositions Using Pairwise Comaximal Ideals

In this article, we show that for a given set of pairwise comaximal ideals {Xi}i∈I in a ring R with unity and any right R-module M with generating set Y and , M = ⊕i∈IC(Xi) if and only if for every y ∈ Y there exists a nonempty finite subset J ⊆ I and positive integers kj such that . We investigate this decomposition for a general class of modules. Our main theorem can be applied to a large class of rings including semilocal rings R with the Jacobson radical of R equal to the prime radical of R, left (or right) perfect rings, piecewise prime rings, and rings with ascending chain condition (ACC) on ideals and satisfying the right AR property on ideals. This decomposition generalizes the decomposition of a torsion abelian group into a direct sum of its p-components. We also develop a torsion theory associated with sets of pairwise comaximal ideals.

On the Cayley Graph of a Commutative Ring with Respect to its Zero-divisors

Let R be a commutative ring with unity and R+ and Z*(R) be the additive group and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) the Cayley graph Cay(R+, Z*(R)). In this article, we study ℂ𝔸𝕐(R). Among other results, it is shown that for every zero-dimensional nonlocal ring R, ℂ𝔸𝕐(R) is a connected graph of diameter 2. Moreover, for a finite ring R, we obtain the vertex connectivity and the edge connectivity of ℂ𝔸𝕐(R). As a result, ℂ𝔸𝕐(R) gives an algebraic construction for vertex transitive graphs of maximum connectivity. In addition, we characterize all zero-dimensional semilocal rings R whose ℂ𝔸𝕐(R) is perfect. We also study Reg(ℂ𝔸𝕐(R)) the induced subgraph on the regular elements of R. This graph gives a family of vertex transitive graphs as well. We show that if R is a Noetherian ring and Reg(ℂ𝔸𝕐(R)) has no infinite clique, then R is finite. Furthermore, for every finite ring R, the clique number and the chromatic number of Reg(ℂ𝔸𝕐(R)) are determined.

Analysis of the magnetically induced current density of molecules consisting of annelated aromatic and antiaromatic hydrocarbon rings.

Magnetically induced current susceptibilities and current pathways have been calculated for molecules consisting of two pentalene groups annelated with a benzene (1) or naphthalene (2) moiety. Current strength susceptibilities have been obtained by numerically integrating separately the diatropic and paratropic contributions to the current flow passing planes through chosen bonds of the molecules. The current density calculations provide novel and unambiguous current pathways for the unusual molecules with annelated aromatic and antiaromatic hydrocarbon moieties. The calculations show that the benzene and naphthalene moieties annelated with two pentalene units as in molecules 1 and 2, respectively, are unexpectedly antiaromatic sustaining only a local paratropic ring current around the ring, whereas a weak diatropic current flows around the C-H moiety of the benzene ring. For 1 and 2, the individual five-membered rings of the pentalenes are antiaromatic and a slightly weaker semilocal paratropic current flows around the two pentalene rings. Molecules 1 and 2 do not sustain any net global ring current. The naphthalene moiety of the molecule consisting of a naphthalene annelated with two pentalene units (3) does not sustain any strong ring current that is typical for naphthalene. Instead, half of the diatropic current passing the naphthalene moiety forms a zig-zag pattern along the C-C bonds of the naphthalene moiety that are not shared with the pentalene moieties and one third of the current continues around the whole molecule partially cancelling the very strong paratropic semilocal ring current of the pentalenes. For molecule 3, the pentalene moieties and the individual five-membered rings of the pentalenes are more antiaromatic than for 1 and 2. The calculated current patterns elucidate why the compounds with formally [4n + 2] π-electrons have unusual aromatic properties violating the Hückel π-electron count rule. The current density calculations also provide valuable information for interpreting the measured (1)H NMR spectra.

Order complex of ideals in a commutative ring with identity

Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere.

On skew cyclic codes over a semi-local ring

In the present paper, we study skew cyclic codes over the finite semi-local ring [Formula: see text], where [Formula: see text] and [Formula: see text] is an odd prime. We define a Gray map from [Formula: see text] to [Formula: see text] and investigate the structural properties of skew cyclic codes over [Formula: see text] using decomposition method. It is proved that the Gray image of a skew cyclic code of length [Formula: see text] over [Formula: see text] is a skew [Formula: see text]-quasi-cyclic code of length [Formula: see text] over [Formula: see text]. Further, it is shown that the skew cyclic codes over [Formula: see text] are principally generated.

Witt's extension theorem for quadratic spaces over semiperfect rings

We prove that every isometry between (not-necessarily orthogonal) summands of a unimodular quadratic space over a semiperfect ring can be extended an isometry of the whole quadratic space. The same result was proved by Reiter for the broader class of semilocal rings, but with certain restrictions on the base modules, which cannot be removed in general.Our result implies that unimodular quadratic spaces over semiperfect rings cancel from orthogonal sums. This improves a cancellation result of Quebbemann, Scharlau and Schulte, which applies to quadratic spaces over hermitian categories. Combining this with other known results yields further cancellation theorems. For instance, we prove cancellation of (1) systems of sesquilinear forms over henselian local rings, and (2) non-unimodular hermitian forms over (arbitrary) valuation rings.Finally, we determine the group generated by the reflections of a unimodular quadratic space over a semiperfect ring.

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3]. Finally, we obtain similar results for the dual class of max modules.