ON SOME SPECIAL CLASSES OF PRIME RINGS

Abstract

Given a non-zero cardinal α, a ring R is said to be SP(α) if a is the first cardinal for which every non-zero element of R has an insulator of cardinality less than α + 1.

It is shown that the class of SP(α) rings is a special class (in the sense of Andrunakievi?) for each α. A theorem of Groenewald and Heyman (also Desale and Varadarajan) to the effect that the class of all strongly prime rings is a special class is obtained as a corollary. Every SP(α) ring has an SP(α) rational extension ring with identity.     

On nonsingular right fpf rings

A right FPF ring is one over which every finitely generated faithful right module is a generator. The main purpose of the article is to givp the following cnaracterization of certain right FPF rings. TheoremLet R be semiprime and right semihereditary. Then R is right FPF iff (1) the right maximal ring of quotients Q_r (R) = Q coincides with the left and right classical rings of quotients and is self-injective regular of bounded index, (ii) R and Q have the same central idem-potents, (iii) if I is an ideal of R generated by a ma­ximal ideal of the boolean algebra of central idempotent s5 R/I is such that each non-zero finitely generated right ideal is a generator (hence prime), and (iv) R is such that every essential right ideal contains an ideal which is essential as a right ideal

In case that R is semiprime and module finite over its centre C, then the above can be used to show that R is FPF (both sides) if and only if it is a semi-hereditary maximal C-order in a self-injective regular ring (of finite index)

In order to prove the above it is shown that for any semiprime right FPF ring R, Q _lcl (R) exists and coincides with Q_r(R) (Faith and Page have shown that the latter is self-injective regular of bounded index). It R is semiprime right FPF and satisfies a polynamical identity then the factor rings as in (iii) above are right FPF and R is the ring of sections of a sheaf of prime right FPF rings

The Proofs use many results of C. Faith and S Page as well as some of the techniques of Pierce sheaves     

ON SOME PROPERTIES OF IF RINGS

Abstract

We show that left IF rings (rings such that every injective left module is flat) have certain regular-like properties. For instance, we prove that every left IF reduced ring is strongly regular. We also give characterizations of (left and right) IF rings. In particular, we show that a ring R is IF if and only if every finitely generated left (and right) ideal is the annihilator of a finite subset of R.     

CLASSICAL QUOTIENT RINGS OF GROUP RINGS

Abstract

Throughout G will denote a free Abelian group and Z(R) the right singular ideal of a ring R. A ring R is a Cl-ring if R is (Goldie) right finite dimensional, R/Z(R) is semiprime, Z(R) is rationally closed, and Z(R) contains no closed uniform right ideals. We prove that R is a Cl-ring if and only if the group ring RG is a C1-ring. If RG has the additional property that bRG is dense whenever b is a right nonzero-divisor, then the complete ring of quotients of RG is a classical ring of quotients.     

HILBERT BASIS QUIVERS

Abstract

A quiver G (= directed multigraph, loops and parallel edges are allowed) is called a Hilbert basis quiver (HBQ) if a certain path algebra R[G] over a ring R is right noetherian provided R does. Such path algebras can be considered as generalized polynomial rings over R. There is the following characterization:

A quiver with a finite number of vertices is HBQ iff its set of edges is finite and its nontrivial path components are elementary cycles, up to parallel edges, which in addition are sink sets (i.e. there is no path leaving the component).

To prove this categorical methods are used.     

TORSION PRERADICALS—A TOOL FOR ANALYSING RINGS

Abstract

This paper surveys a selection of results in the literature on torsion preradicals; these are left exact preradical functors on the category of unital right modules over an associative ring with identity. Various well known classes of rings such as semisimple, artinian, perfect and strongly prime are characterized in terms of torsion preradicals. A classification of prime rings using torsion preradicals is also exhibited. Rings all of whose torsion preradicals are radicals and rings whose torsion preradicals commute, are investigated. An application of the latter condition to Jacobson's Conjecture is presented.     

UNIFORMLY STRONGLY PRIME RADICALS

Abstract

In this article different characterizations for a uniformly strongly prime ring are given as well as a way of constructing a uniformly strongly prime ring. Uniformly strongly prime rings of bound one as well as the upper radical determined by this special class of rings are also investigated.     

ASSOCIATIONS AND VALUATIONS

Abstract

Right cones are semigroups for which the lattice of right ideals is a chain and a left cancellation law holds; valuation rings, the cones of ordered groups, and initial segments of ordinal numbers are examples. Two such cones are associated if they have isoniorphic lattices of right ideals so that ideals, prime ideals, and completely prime ideals correspond to each other. A list of problems is discussed. In Proposition 3.11 it is proved that the canonical mapping from a right invariant right chain domain R onto the associated right holoid can be extended to a valuation from the skew field Q(R) of quotients of R onto an ordered group if and only if Ja ? aJ for all a ∈ R and J = J(R), the Jacobson radical of R.     

K-Theory of Free Rings

The object of this article is to establish the following result (Corollary 3.9 below): If R is a regular right noetherian ring and R{X} is the free associative algebra on the set X, then K_n(R) = K_n(R{X}), where K_n refers to the Quillen K-theory. The result can be stated in the equivalent form that H_n(G1(R),Z) = H_n(G1(R{X}),Z). From this result it follows that if F is a free ring without unit, then K_n(F) = 0, whence free rings are acyclic models for Quillen K-theory (3.11 below). This result in turn enables us to complete Anderson's work [1] in identifying the Quillen K-theory [11] and the K-theory proposed by Gersten [7] and Swan [18] for all rings. We also establish that the natural transformation K_n(R) → K_n ^k-v(R) between the Quillen theory and the K-theory of Karoubi and Villamayor is an isomorphism if R is a supercoherent (Definition 1.2) and regular (Definition 1.3) ring. From this result we can gain some information about the K-theory of group rings of free products of groups (Theorem 5.1).     

幺半群上的斜强可逆环

For a monoid M, we introduce the concept of skew strongly M-reversible rings which is a generalization of strongly M-reversible rings, and investigate their properties. It is shown that if G is a finitely generated Abelian group, then G is torsion-free if and only if there exists a ring R with |R| ≥ 2 such that R is skew strongly G-reversible. Moreover, we prove that if R is a right Ore ring with classical right quotient ring Q, then R is skew strongly M-reversible if and only if Q is skew strongly M-reversible.     

Weakly regular rings with ACC on annihilators and maximality of strongly prime ideals of weakly regular rings

Ramamurthi proved that weak regularity is equivalent to regularity and biregularity for left Artinian rings. We observe this result under a generalized condition. For a ring R satisfying the ACC on right annihilators, we actually prove that if R is left weakly regular then R is biregular, and that R is left weakly regular if and only if R is a direct sum of a finite number of simple rings. Next we study maximality of strongly prime ideals, showing that a reduced ring R is weakly regular if and only if R is left weakly regular if and only if R is left weakly π-regular if and only if every strongly prime ideal of R is maximal.     

强symmetric环

为了统一交换环和约化环的层表示,Lambek引进了Symmetric环.继续symmetric环的研究,定义引入了强symmetric环的概念,研究它的一些扩张性质.证明环R是强symmetric环当且仅当R[x]是强symmetric环当且仅当R[x;x~(-1)]是强symmetric环.也证明对于右Ore环R的经典右商环Q,R是强symmetric环当且仅当Q是强symmetric环.     

Weakly Special Classes of Semiprime Rings

This paper investigates closure properties possessed by certain classes of finite subdirect products of prime rings. If ℳ is a special class of prime rings then the class ℳ_∞ of all finite subdirect products of rings in ℳ is shown to be weakly special. A ring S is said to be a right tight extension [resp. tight extension] of a subring R if every nonzero right ideal [resp. right ideal and left ideal] of S meets R nontrivially. Every hereditary class of semiprime rings closed under tight extensions is weakly special. Each of the following conditions imposed on a semiprime ring yields a hereditary class closed under right tight extensions: ACC on right annihilators; finite right Goldie dimension; right Goldie. The class of all finite subdirect products of uniformly strongly prime rings is shown to be closed under tight extensions, answering a published question. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Matrix Rings and Subhereditary Radicals

Abstract

We study some one-sided ideals of row bounded and of column bounded matrix rings over free algebras. Obtained results are applied to answer several open problems on subhereditary radicals of associative rings. In particular we show that the right strongly prime radical is not left subhereditary and that the lattice of left (right) subhereditary radicals is not complete.     

TRANSCENDENTAL AND ALGEBRAIC EXTENSIONS OF COMMUTATIVE RINGS

Abstract

Transcendental and algebraic elements over commutative rings are defined. Rings with zero nil radical are considered. For a transcendental over R, necessary and sufficient conditions are derived for elements of R[α] to be algebraic or transcendental over R. For R a ring with identity and a finite number of minimal prime ideals, necessary and sufficient conditions are given for any element in a unitary overring of R to be algebraic or transcendental over R. It is proved that if α is algebraic Over R, so is every element of R[α]. It is show that if R is Noetherian, β is algebraic over R[α] and α is algebraic over R, then, under certain conditions, β is algebraic over R. If R has a finite number of minimal prime ideals, P₁,…,P_k, which are pairwise comaximal, then if t is transcendental over R, R[t] can be obtained by adjoining k algebraic elements a_i over R to R whose defining polynomials are in P_i [x], and conversely, if such elements are adjoined to R, they generate an element transcendental over R.     

On Coprime Modules and Comodules

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory.

Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.     

PRIME IDEALS IN RINGS WITH INVOLUTION

Abstract

Let R be a ring with involution *. We show that if R is a *-prime ring which is not a prime ring, then R is “essentially” a direct product of two prime rings. Moreover, if P is a *-prime *-ideal of R, which is not a prime ideal of R, and X is minimal among prime ideals of R containing P, then P is a prime ideal of X, P = X ∩ X* and either: (1) P is essential in X and X is essential in R; or (2) for any relative complement C of P in X, then C* is a relative complement of X in R. Further characterizations of *-primeness are provided.     

Group graded rings

Let R be a group graded ring . The map ( , ): R × R →₁ defined by: (x,y) = (xy)₁ , is an inner product on R. In this paper we investigate aspects of nondegeneracy of the product, which is a generalization of the notion of strongly G —graded rings,introduced by Dade. We show that various chain conditions are satisfied by R if and only if they are satisfied by R₁ , and that when R₁ is simple artinian, then R is a crossed product R₁ ^* G. We give conditions for simple R-modules to be completely reducible R₁ -modules . Finally, we prove an incomparability theorem,when G is finite abelian.     

Uniserial Noetherian centrally essential rings

Abstract

It is proved that a ring R is a right uniserial, right Noetherian centrally essential ring if and only if R is a commutative discrete valuation domain or a left and right Artinian, left and right uniserial ring. It is also proved that there exist non-commutative uniserial Artinian centrally essential rings.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.