Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.     

Ideals of Some Matrix Rings

For a commutative ring K the conception of a strongly maximal ideal J was introduced by Kuzucuoglu and Levchuk in 2000. Denote by R_n(K,J) the ring of all n×n-matrices over K with elements from J on and above the main diagonal. Recent results on ideals of the ring R_n(K,J) for this case, ideals of the associated Lie ring and normal subgroups of the adjoint group are considered in this paper. Also ideals of R_n(K,J) for the case of an arbitrary associative ring K with the identity are investigated.     

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.     

A Class of Quasipolar Rings

An element a of a ring R is called J-quasipolar if there exists p ² = p ∈ R satisfying p ∈ comm²(a) and a + p ∈ J(R); R is called J-quasipolar in case each of its elements is J-quasipolar. The class of this sort of rings lies properly between the class of uniquely clean rings and the class of quasipolar rings. In particular, every J-quasipolar element in a ring is quasipolar. It is shown, in this paper, that a ring R is J-quasipolar iff R/J(R) is boolean and R is quasipolar. For a local ring R, we prove that every n × n upper triangular matrix ring over R is J-quasipolar iff R is uniquely bleached and R/J(R) ? ?₂. Moreover, it is proved that any matrix ring of size greater than 1 is never J-quasipolar. Consequently, we determine when a 2 × 2 matrix over a commutative local ring is J-quasipolar. A criterion in terms of solvability of the characteristic equation is obtained for such a matrix to be J-quasipolar.     

The structure of a class of Z-local rings

A local ring R is called Z-local if J(R) = Z(R) and J(R)² = 0. In this paper the structure of a class of Z-local rings is determined.     

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.     

When is a Sum of Annihilator Ideals an Annihilator Ideal?

We call a ring R a right SA-ring if for any ideals I and J of R there is an ideal K of R such that r(I) + r(J) = r(K). This class of rings is exactly the class of rings for which the lattice of right annihilator ideals is a sublattice of the lattice of ideals. The class of right SA-rings includes all quasi-Baer (hence all Baer) rings and all right IN-rings (hence all right selfinjective rings). This class is closed under direct products, full and upper triangular matrix rings, certain polynomial rings, and two-sided rings of quotients. The right SA-ring property is a Morita invariant. For a semiprime ring R, it is shown that R is a right SA-ring if and only if R is a quasi-Baer ring if and only if r(I) + r(J) = r(I ∩ J) for all ideals I and J of R if and only if Spec(R) is extremally disconnected. Examples are provided to illustrate and delimit our results.     

Ring elements as sums of units

In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.      

Normal Structure of the Adjoint Group in the Radical Rings Rn(K,J)

The main purpose of this article consists in investigating the conjecture of existence of an algorithm for constructing normal subgroups of the adjoint group of the ring R _n (K,J) from its Lie ideals under natural restrictions on K and J.     

On JB -Rings

A ring R is a QB-ring provided that aR + bR = R with a, b ∈ R implies that there exists a y ∈ R such that It is said that a ring R is a JB-ring provided that R/J(R) is a QB-ring, where J(R) is the Jacobson radical of R. In this paper, various necessary and sufficient conditions, under which a ring is a JB-ring, are established. It is proved that JB-rings can be characterized by pseudo-similarity. Furthermore, the author proves that R is a JB-ring iff so is R/J(R)².     

Presimplifiable and Cyclic Group Rings

Let R be any commutative ring with identity, and let C be a (finite or infinite) cyclic group. We show that the group ring R(C) is presimplifiable if and only if its augmentation ideal I(C) is presimplifiable. We conjecture that the group rings R(C _n ) are presimplifiable if and only if n = p ^m , p ∈ J(R), p is prime, and R is presimplifiable. We show the necessity of n = p ^m , and we prove the sufficiency when n = 2, 3, 4. These results were made possible by a new formula derived herein for the circulant determinantal coefficients.     

The structure of symplectic groups over arbitrary commutative rings

LetR be an arbitrary commutative ring, andn be an integer ≥3. It is proved for any idealJ ofR thatESp _2n(R,J)=[ESp _2n(R),ESp _2n(J)]=[ESp _2n(R),ESp _2n(R,J)] =[ESp _2n(R),GSp _2n(R,J)]=[Sp _2n(R),ESp _2n(R,J)]. Furthermore, the problem of normal subgroups ofSp _2n(R) has an affirmative solution if and only ifaR=a ²R+2aR for eacha inR. This covers the relevant results of [4], [8], [10], [12] and [13]. Project Supported by the Science Fund of the Chinese Academy of Sciences     

FINITE RINGS WITH COMMUTING NILPOTENT ELEMENTS

As a generalization of Wedderburn's theorem, Herstein [5] proved that a finite ring R is commutative, if all nilpotent elements are contained in the center of R. However a finite ring with commuting nilpotent elements is not necessarily commutative. Recently, in [9] and [10], Simons described the structure of finite rings R with J(R)² = 0 in a variety with definable principal congruences. In this paper, we will consider the difference between the finite commutative rings and the finite rings in which any two nilpotent elements commute with each other. As a consequence, we describe the structure of finite rings R with [J(R), J(R)] = 0 in a variety with definable principal congruences.     

Jacobson radical of skew polynomial rings and skew group rings

LetR be a ring and σ an automorphism ofR. We prove the following results: (i)J(R _σ[x])={Σ_ir_i x ⁱ:r₀∈I∩J(R]), r _i∈I for alliε 1} whereI↪ {r∈R:rx ∈J(R _Σ[x])|s= (ii)J(R _σ<x>)=(J(R _σ<x>)∩R)_σ<x>. As an application of the second result we prove that ifG is a solvable group such thatG andR, + have disjoint torsions thenJ(R)=0 impliesJ(R(G))=0.     

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ? J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.     

Finitistic dimensions of semiprimary rings

LetR be a semiprimary ring. We show that if the left generalized projective dimension (defined below) of _R (R/J ²) is finite, then the injectively defined left finitistic dimension ofR is finite.     

On a graph of ideals

We associate a graph Γ₊(R) to a ring R whose vertices are nonzero proper right ideals of R and two vertices I and J are adjacent if I+J=R. Then we try to translate properties of this graph into algebraic properties of R and vice versa. For example, we characterize rings R for which Γ₊(R) respectively is connected, complete, planar, complemented or a forest. Also we find the dominating number of Γ₊(R).     

α-Baer rings and some related concepts via C(X)

We call a commutative ring R an F IN -ring (resp., F SA-ring) if for any two finitely generated I, J ?R we have Ann(I)+Ann(J )=Ann(I∩J ) (resp., there is K ? R such that Ann(I)+Ann(J )=Ann(K)). Moreover, we extend this concepts to αIN -rings and αSA-rings where α is a cardinal number. The class of F SA-rings includes the class of all SA-rings (hence all IN -rings) and all P P -rings (hence all Baer-rings). In this paper, after giving some properties of αSA-rings, we prove that a reduced ring R is αSA if and only if it is an αIN -ring. Consequently, C(X) is an F SA-ring if and only if C(X) is an F IN -ring and equivalently X is an F -space. Moreover, for a commutative ring R, we have shown that R is a Baer-ring if and only if R is a reduced IN -ring. A topological space X is said to be an αU E-space if the closure of any union with cardinal number less than α of clopen subsets is open. Topological properties of αU E-spaces are investigated. Finally, we show that a completely regular Hausdor? space X is an αU E-space if and only if C(X) is an αEGE-ring.     

On Strongly J-Clean Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J-clean in case each of its elements is strongly J-clean. We investigate, in this article, strongly J-clean rings and ultimately deduce strong J-cleanness of T _n (R) for a large class of local rings R. Further, we prove that the ring of all 2 × 2 matrices over commutative local rings is not strongly J-clean. For local rings, we get criteria on strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices. The strong J-cleanness of a 2 × 2 matrix over commutative local rings is completely characterized by means of a quadratic equation.     

Group Identities and Prime Rings Generated by Units

Abstract

Let R be a prime ring and U(R) the group of units of R. We prove that if U(R) generates R and satisfies a group identity,then R is either a domain or a full matrix ring over a finite field.