ON CERTAIN NEW RADICALS ARISING FROM A GIVEN RADICAL

Olson and Jenkins defined E(M) to be the class of all rings each nonzero homomorphic image of which contains either a nonzero M-ideal or an essential ideal where M, is any class of rings. E(M) was proven to be a radical class and various classes M were considered. Here the class E(M) is partitioned into two classes: H the class of all rings each nonzero homomorphic image of which has a proper essential ideal and the class H(M) of all rings each nonzero homomorphic image of which contains an M-ideal. It is shown that H is a radical class and under certain conditions H(M) is also a radical class. Various properties placed on M yield several well-known radical classes and an infinite number of supernilpotent nonspecial radical classes is constructed.

Conditions are given on a radical (semisimple) property to ensure the existence of a construction for the smallest radical (semisimple) class with the property, containing a given class of rings. This generalizes earlier results on smallest hereditary or strongly hereditary radical classes, and hereditary semisimple classes. In the last section certain classes of rings are shown to admit a construction for the largest radical (semisimple) class contained in the given class. This leads to theorems on largest radical (semisimple) classes dual to the already established smallest theorems.

Let A be a class of rings, and let L(A) denote the lower radical class determined by A. In [3] Yu-Lee Lee showed that L(A) may b constructed in the following manner: Let H(A) be the class of all homomorphic images of rings in A.

Some Pathology for Radicals in Non-Associative Near-Rings

On the Schröder–Bernstein property for modules

A ring R for which x2 = 0 for all x e R is called a zerosquare ring. Zero-square rings are easily seen to be locally nilpotent. This leads to two problems: (1) constructing finitely generated zero-square rings with large index of nilpotence, and (2) investigating the structure of finitely generated zerosquare rings with given index of nilpotence. For the first problem we construct a class of zero-square rings, called free zero-square rings, whose index of nilpotence can be arbitrarily large. We show that every zero-square ring whose generators have (additive) orders dividing the orders of the generators of some free zero-square ring is a homomorphic image of the free ring. For the second problem, we assume Rn Φ 0 and obtain conditions on the additive group R+ of R (and thus also on the order of R). When n = 2, we completely characterize R+. When n > 3 we obtain the smallest possible number of generators of R+, and the smallest number of generators of order 2 in a minimal set of generators. We also determine the possible orders of R. Trivially every null ring (that is, R2 = 0) is a zero-square ring. From every nonnull commutative ring S we can make S x S x S into a nonnull zero square ring R by defining addition componentwise and multiplication by (x19 ylf Zj) x (x2, y2, z2) = (0, 0, x,y2 - x2y,) .

The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:- (a) To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc. (b) To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules. (c) To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules. (d) To examine the structure of certain classes of abelian groups and to try to find modules with similar structures. (e) To find roles of answers to some of above types of questions in the general theory of modules.

In this paper we propose a general setting in which to study the radical theory of group graded rings. If is a radical class of associative rings we consider two associated radical classes of graded rings which are denoted by G and ref. We show that if is special (respectively, normal), then both G and ref are graded special (respectively, graded normal). Also, we discuss a graded version of the ADS theorem and the termination of the Kurosh lower graded radical construction.

Regarding the construction of strict and base radical classes for the variety of associative rings, we show that “subring” and “accessible subring” are two of a number of substructures that can be used in constructing a radical class. The substructures we focus on are set properties that are transitive across accessible subrings. These substructures allow us to look beyond those that are related by the binary operations of the ring itself, to ones that may only be related by containment. The lower radical type construction defined by these substructures include those of Yu-lee Lee, McDougall and Stewart.

Let T be an abelian group and λ an uncountable regular cardinal. The question of whether there is a λ-universal group U among all torsion-free abelian groups G of cardinality less than or equal to λ satisfying Exta(G,T) = 0 is considered. U is said to be λ-universal for T if, whenever a torsion-free abelian group G of cardinality at most λ satisfies Ext(G,T)=0, there is an embedding of G into U. For large classes of abelian groups T and cardinals λ, it is shown that the answer is consistently no, that is to say, there is a model of ZFC in which, for pairs T and λ, there is no universal group. In particular, for T torsion, this solves a problem by Kulikov.

In part I we introduce vertex rings , which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over C \mathbf {C} . We show that vertex rings are characterized by Goddard axioms . These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the rôle played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced étale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.

Nonlocalness of a class of lattice-orderable associative algebras and rings

It is shown that a variety of associative rings which has attainable identities in the class of all associatives ring has attainable identities in the class of all near—rings. We also give examples of varieties of near-rings which are, contrary to the ring case, closed under extensions but do not have attainable identities and varieties which are closed under extensions but not closed under essential extensions, respectively.

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

Let \({\sigma}\) be the class of all associative rings R with the property that if I and J are ideals of R and for all \({i \in I, j \in J}\) there exist natural numbers n, m such that i n j m = 0, then I = 0 or J = 0. We will show that \({\sigma}\) is a special class and the upper radical \({\mathcal{U}(\sigma)}\) determined by \({\sigma}\) is equal to the nil radical \({\mathcal{N}}\) of Koethe. This implies that the nil radical \({\mathcal{N}(R)}\) of any ring R is the intersection of all \({\sigma}\)-ideals I of R, that is, ideals I of R such that the factor ring R/I is in \({\sigma}\). We also give a necessary and sufficient condition for an ideal \({I \neq R}\) of a ring R to be a \({\sigma}\)-ideal.