ON THE STRUCTURE OF THE JACOBSON RADICAL OF GRADED RINGS

Abstract

We introduce a new large class of semigroups S including all locally finite, completely regular and strongly π-regular linear semigroups. For any semigroup S in the class and any S-graded ring R, the structure of the Jacobson radical of R is reduced to the radicals of subrings graded by the maximal subgroups of S. Many results on radicals follow from this reduction in a unified way. In two special cases the reduction is simplified.     

Kurosh–Amitsur Right Jacobson Radicals of Type-1 and 2 for Right Near-Rings

Near-rings considered are right near-rings. Let ν ∈ {1, 2}. J ^r _ν , the right Jacobson radical of type-ν, was introduced for near-rings by the first and second authors. In this paper properties of these radicals J ^r _ν are studied. It is shown that J ^r _ν is a Kurosh-Amitsur radical (KA-radical) in the variety of all near-rings R in which the constant part R _c of R is an ideal of R. Thus, unlike the left Jacobson radical of type-1 of near-rings, J ^r ₁ is a KA-radical in the class of all zero-symmetric near-rings. J ^r _ν is not s-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings. Received: April 1, 2007. Revised: July 11, 2007.     

Jacobson rings and rings strongly graded by an abelian group

LetR be ring strongly graded by an abelian groupG of finite torsion-free rank. Lete be the identity ofG, andR _e the component of degreee ofR. AssumeR _e is a Jacobson ring. We prove that graded subrings ofR are again Jacobson rings if eitherR _e is a left Noetherian ring orR is a group ring. In particular we generalise Goldie and Michlers’s result on Jacobson polycyclic group rings, and Gilmer’s result on Jacobson commutative semigroup rings of finite torsion-free rank.     

Semilocal rings whose adjoint group is locally supersoluble

An associative ring R, not necessarily with an identity element, is called semilocal if R modulo its Jacobson radical is an artinian ring. It is proved that if the adjoint group of a semilocal ring R is locally supersoluble, then R is locally Lie-supersoluble and its Jacobson radical is contained in a locally Lie-nilpotent ideal of finite index in R.     

Commutative Rings R Whose C(𝔸𝔾(R)) Consist of Only Triangles

Let R be a commutative ring with identity. The set 𝕀(R) of all ideals of R is a bounded semiring with respect to ordinary addition, multiplication and inclusion of ideals. The zero-divisor graph of 𝕀(R) is called the annihilating-ideal graph of R, denoted by 𝔸𝔾(R). We write 𝒢 for the set of graphs whose cores consist of only triangles. In this paper, the types of the graphs in 𝒢 that can be realized as either the zero-divisor graphs of bounded semirings or the annihilating-ideal graphs of commutative rings are determined. A necessary and sufficient condition for a ring R such that 𝔸𝔾(R) ∈ 𝒢 is given. Finally, a complete characterization in terms of quotients of polynomial rings is established for finite rings R with 𝔸𝔾(R) ∈ 𝒢. Also, a connection between finite rings and their corresponding graphs is realized.     

On V-semirings and Semirings All of Whose Cyclic Semimodules Are Injective

In this article, we introduce and study V- and CI-semirings—semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and antibounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple antibounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.     

RADICAL THEORY FOR SEMIRINGS

Abstract

In this article we continue investigations on a Kurosh-Amitsur radical theory for a universal class U of hemirings as introduced by O.M. Olson et al. We give some necessary and sufficient conditions that such a universal class U consists of all hemirings. Further we consider special and weakly special subclasses M of U which yield hereditary radical classes P = um of U. In this context we correct some statements in the papers of Olson et al. Moreover, a problem posed there concerning the equality of two radicals ?_∞(S) and ?^ε(S) and two similar ideals β_∞ (S) and β^ε(S) is widely solved. We prove ?_∞(S) ? ?^ε(S) = β^∞(S) = β^ε(S) and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class M_ε(U) is always semisimple, a result which is not true for the special class M_∞(U).     

Semirings of Formal Power Series

Semirings of formal power series over semirings, in particular, over k-semifields, are studied. It is also shown that the semiring of formal power series S[[x]] over a k-semifield S becomes a local semiring. Moreover, the Jacobson radical of S[x]] over a k-semifield S is described.AMS Subject Classification (1991): Primary 16Y60     

Addendum to “examples of semiperfect rings”

We fill a gap in [4], and provide a rigorous example of a local ringR whose Jacobson radical is locally nilpotent, butM ₂(R) is not strongly π-regular. The online version of the original article can be found at The work of the first author was partially supported by the DGICYT (Spain), through the grant PB92-0586, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya.     

The Semigroup Structure of Left Clifford Semirings

In this paper,we generalize Clifford semirings to left Clifford semirings by means of the so-called band semirings.We also discuss a special case of this kind of semirings,that is, strong distributive lattices of left rings.     

Gelfand Factor Rings and Weak Zariski Topologies

Let R be any ring with identity. Let N(R) (resp. J(R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec _r (R) (resp. Spec _l (R), Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec _r (R) (with weak Zariski topology). The following results are obtained: (1) R/N(R) is a Gelfand ring if and only if Spec _r (R) is a normal space if and only if Spec _l (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.     

Lifting Central Idempotents Modulo Jacobson Radicals and Ranks of K 0 Groups

Let R be a ring and let J(R) be the Jacobson radical of R. We discuss the problem of determining when the central idempotents in R/J(R) can be lifted to R. If R is a noetherian (artinian) ring, we give some conditions relative to the ranks of K ₀ groups under which the central idempotents in R/J(R) can be lifted. In particular, when R is semilocal, these conditions are necessary and sufficient. Moreover, we consider ranks of K ₀ groups of pullbacks of rings and obtain the upper and lower bounds on them under some suitable conditions.     

Jordan radicals of u-algebras

In this paper we show that strong noncommutative Jordan algebras R over an arbitrary ring of scalars having the alternator mappings y,y,-1 as Jordon derivations are U-algebras, algebras such that U_ablpar;crpar; lies in the Jordan ideal generated by a. For any U-algebra R we relate the radical theories of R and R⁺. Our main result is that any radical property p′ of U-algebras such that P′-radR? p-radR⁺. If p is nondegenerate the P′ is nondegenerate and P′-radR=p-radR⁺. This applies in particular to the McCrimmon, locally nilpotent, nil, Jacobson and Brown-McCoy radicals of Jordan algebras     

Radicals of Jordan rings connected with alternative rings

Subject to a certain restriction on the additive group of an alternative ring A, we prove that R(A)=R(A⁽⁺⁾), where A⁽⁺⁾ is a Jordan ring and R is one of the following radicals: the Jacobson radical, the upper nil-radical, the locally nilpotent radical, or the lower nil-radical. For the proof of these relationships Herstein's well-known construction for associative rings is generalized to alternative rings.     

Toward Homological Characterization of Semirings: Serre’s Conjecture and Bass’s Perfectness in a Semiring Context

Among other results on homological characterization of semirings, we prove that the classes of projective and free right (left) semimodules over the polynomial semiring R[x₁, x₂,..., x_n] over an additively regular division semiring R coincide iff R is a field. Also it is shown that an additively regular commutative semiring R is perfect (in H. Basss sense) iff R is a perfect ring.In Celebration of the Sixtieth Birthday of Ralph N. McKenzieReceived July 27, 2003; accepted in final form April 2, 2004.     

ALMOST COHEN–MACAULAY MODULES

A commutative noetherian ring R is called an almost Cohen–Macaulay ring if depth(P, R) = depth(P R _P , R _P ) for every P ∈ SpecR. [It was called a D-ring by Han (Acta Math. Sinica 1998, 4, 1047–1052).] Several fundamental properties of almost Cohen–Macaulay rings were estabished by Han. In this note, a new characterization is proved: R is an almost Cohen–Macaulay ring if and only if height P ≤ 1 + depth(P, R) for every P ∈ Spec(R). By this characterization, we settle an unsolved problem in Han's paper: R is an almost Cohen–Macaulay ring if and only if so is the power series ring R[[X ₁, …, X _n ]]. The notion of an almost Cohen–Macaulay ring is generalized to that of an almost Cohen–Macaulay module in this note.     

The Von Neumann Regular Radical and Jacobson Radical of Crossed Products

We construct the H-von Neumann regular radical for H-module algebras and show that it is an H-radical property. We obtain that the Jacobson radical of a twisted graded algebra is a graded ideal. For a twisted H-module algebra R, we also show that r _j (R# H) = r _Hj (R)# H and the Jacobson radical of R is stable, when k is an algebraically closed field or there exists an algebraic closure F of k such that r _j (RF) = r _j (R) F, where H is a finite-dimensional, semisimple, cosemisimple, commutative or cocommutative Hopf algebra over k. In particular, we answer two questions of J. R. Fisher.