A NOTE ON THE g-PRIME RADICAL IN GAMMA RINGS

ABSTRACT

The g-prime radical of a Γ-ring M is equal to either the zero ideal or the prime radical of M. If the prime radical of M is a non-zero ideal, then the following three conditions are equivalent; (i) g-prime radical of M is equal to the prime radical of M; (ii) every g-prime ideal is a prime ideal; and (iii) every g-semiprime ideal is a semiprime ideal.     

CHARACTERIZING THE NIL RADICAL OF A GAMMA RING

ABSTRACT

The nil radical, N(M) of a Γ-ring M was defined by Coppage and Luh [3], and shown by Groenewald [4] to be a special radical. We define s-prime ideals of M and show that N(M) is equal to the intersection of the s-prime ideals of M. If R is a ring, the nil radical of R considered as a Γ-ring with Γ = R is equal to the upper nil radical of R. We also give a sufficient condition for the equality N(R)* = N(M), where R is the right operator ring of M, and N(R) is its upper nil radical.     

THE ADDITIVE GROUPS OF RINGS WITH TOTALLY ORDERED LATTICE OF IDEALS

The object of this paper is to describe the additive groups of rings R satisfying the following condition: For any two ideals A,B in R either A u B, or B u A. Such rings will be called TOLI rings. The rings considered here are not necessarily associative or with identity.     

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 THE GROENEWALD-HEYMAN STRONGLY PRIME RADICAL

Strongly prime rings were introduced by Handelman and Lawrence [6], and in a recent paper [5] Groenewald and Heyman investigated the upper radical determined by the class of all strongly prime rings. In this paper we continue this investigation. Section 1 provides some alternative characterizations of the radical and in section 2 we discuss general properties of the radical and compare it with other well-known radicals. Finally, combinatorial results on polynomial identities are presented which, combined with our results in section 2. yield some new comnutativity theorems.

All rings considered are associative, but do not necessarily have an identity. As usual, I Δ A means that I is an ideal of the ring A. The notation <x₁,x₂,…> and (x_l,x₂,…) will stand for the subring and ideal, respectively, generated by the elements x₁,x₂,…. The rignt annihirator of a subset S of a ring A will be denoted by ann_A(S).

This work was supported in part by NSERC grants A-8775 and A-8789. and was completed while the first and third authors were visiting Dalhousie University. These authors would like to thank the Department of Mathematics, Statistics and Computing Science at Oalhousie University for its generous hospitality.     

ON ANTISIMPLE GAMMA RINGS

ABSTRACT

In this note, we define the antisimple radical, A(M), of a Γ-ring M. A(M) is shown to be a special radical, and two characterizations of antisimple rings due to Szész are extended to Γ-rings. If R is the right operator ring of M, then A(R)* = A(M), where A(R) is the antisimple radical of R. If m,n are positive integers, then A(M_mn) = (A(M))_mn, where M_mn denotes the group m x n matrices over M, considered as a Γ_nm -ring with the operations of matrix addition and multiplication.     

BOUNDS FOR INDICES OF NILPOTENCY OF FINITELY GENERATED NIL RINGS OF FINITE INDEX

Abstract

An improved bound is given for the index of nil-potency of a finitely generated nil ring of index n in terms of the index of nilpotency of the ideal generated by T^m where m = [n/2] and T is a m-subset of the set of generators. If m = 3 it is proved that T¹⁰ is contained in an ideal generated by twenty-seven cubes and this is applied to get bounds for the index of nilpotency of a finitely generated nil ring of index 6 or 7, bounds which are less than one hundredth of the bounds we obtained in a previous paper.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

ON A CLASSIFICATION OF PRIME RINGS

Abstract

We formalize the adjunction of maps of localized nilpotent spaces of the homotopy type of CW-complexes. Using the semilocalization theory of M. Bendersky, in which only the higher homotopy groups are localized, we obtain an adjunction theorem with fewer nilpotency conditions on the spaces. The two main theorems are in the form of a theorem on maps of triads by J. P. May.     

*-RINGS AND THEIR RADICALS

ABSTRACT

This paper is to investigate the class * of all prima rings without proper non-zero prima factors and to characterize the ‘minimal special radical’ containing the class * as well as the ‘maximal special radical’ the semisimple class of which contains the class *. This let us answer certain open questions put in [1] and [4].     

ALHOST M-RINGS AND M*-RINGS

ABSTRACT

For a class C of rings, Olson in 1982, and Le Roux and Heyman in 1980, introduced the definitions of classes C ₁ and C* of rings respectively. The aim of this paper is to show the equivalence of these two definitions if C satisfies certain properties. Furthermore the relation between C ₁ and C*, and UC and UC* will be investigated if C does not necessarily satisfy these restricted properties.     

REGULARITIES AND SUBDIRECT PRODUCTS OF RINGS

Abstract

Structure theorems are obtained for certain radical classes of rings (including the Brown-McCoy radical class, the class of λ-rings, the class of E ₅-rings, the class of E ₆-rings and the class of f-regular rings) by generalizing the concept of a prime ideal.     

PRIME AND SEMIPRIME BI-IDEALS

Abstract

Prime and semiprime bi-ideals in associative rings are defined. This provides a setting for a generalization of the well-known theorem that a commutative ring is Von Neumann regular iff every ideal is semiprime.     

A NOTE ON FIXED POINTS OF CERTAIN RELATIVELY FREE NILPOTENT GROUPS

Let P be the variety of all polynilpotent groups of class row (c ₁,…,c_r ) and N _c be the variety of all nilpotent groups of class c. In this note, we give a necessary and sufficient condition for a relatively free group F_n (P ∧ N _c ) of finite rank n, with n ≥ 2, to have a non-trivial element fixed by every automorphism of F_n (P ∧ N _c ). Using this condition and the works of Wever [6] Wever, F. 1947. Über Invarianten in Lie'schen Ringen. Math. Ann., 120: 44–55.  [Google Scholar] and Burrow ([1] Burrow, M. 1958. Invariants of Free Lie Rings. Commun. Pure Applied Maths, XI: 4197–431.  [Google Scholar] and [2] Burrow, M. 1967. The Enumeration of Lie Invariants. Commun. Pure Applied Maths, XX: 401–411.  [Google Scholar]), we give a full answer to a Miasnikov's question.     

A matrix problem over a central quadratic skew field extension

Let F⊂G=F(u) be a central quadratic skew field extension (such that the generator u is central in G) and a natural (G,G)-bimodule. We deal with the matrix problem on finding a canonical form for rectangular matrices over W with help of left elementary transformations of their rows and right elementary transformations of columns over G. We solve this problem reducing it in the separable (resp. inseparable) case to the semilinear (resp. pseudolinear) pencil problem.     

A NOTE ON REGULAR AND EXTREYAL SUBOBJECTS

ABSTRACT

In the general setting of a complete, well-powered category A, we define and study two universal closure operations: regularization and extremalization, by means of regular and extremal subobjects of A. respectively. A general theorem of characterization of epimorphisms in A is given. When A is an epireflective subcategory of TOP, such operations are shown to coincide with A-closure [11] and epiclosure [2]. respectively. In the topological contest, regularization and extremalization are studied in detail and compared with r-closure, defined in [13].     

NOTES ON NILPOTENCY OF NIL SUBRINGS OF ENDOMORPHISM RINGS OF MODULES

Abstract

A family K of right R-modules is called a natural class if K is closed under submodules, direct sums, infective hulls, and isomorphic copies. The main result of this note is the following: Let K be a natural class on Mod-R and M ε K. If M satisfies a.c.c. (or d.c.c.) on the set of submodules {N ? M: M/N ε K}, then each nil subring of End(M_R ) is nilpotent.