THE BROWN-MCCOY RADICAL AND ONE-SIDED IDEALS

Abstract

Some strong and one-sided hereditary radicals connected with the Brown-McCoy radical are studied.     

Differential rings and ore extensions: Brown-McCoy rings

We consider here a ringK, a derivationD ofK and the differential polynomial ringR=K[X;D]. The ringK is said to be a Brown-McCoy ring if the prime radical coincides with the Brown-McCoy radical in every homomorphic image ofK. AD-Brown-McCoy ring is defined in a similar way. We prove the following conditions are equivalent: (i)K is aD-Brown-McCoy ring; (ii)R is a Brown-McCoy ring and for every maximal idealM ofR,K/(MνK) is aD-simple ring with 1. In addition, we give some applications and examples on the study of the transfer of the property of being a Brown-McCoy ring betweenK andR. Further, we study the relation between the prime and theD-prime ideals of a differential intermediate extension of a liberal extension. This paper was supported by a fellowship awarded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.     

SUBDIRECT IRREDUCIBILITY AND RADICALS

Abstract

We prove lemmas in Andrunakievich s-varieties on the transitivity of the relation “is an ideal of” and concerning subdirectly irreducible factor rings. Applying these lemmas we show that a Plotkin radical introduced in [8] has the ADS-property and is ideal hereditary. These lemmas are applicable in proving a subdirect decomposition for rings having an ideal with 0 antisimple radical. For Jordan algebras and near-rings (they do not form Andrunakievich varieties) we can prove a similar subdirect decomposition concerning ideals with 0 Brown-McCoy radical.     

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.     

分次广义Γ-环的Brown-McCoy根

研究了分次广义Γ-环的弱强分次B row n-M cC oy根与拟弱强分次B row n-M cC oy根,并从不同角度刻划了分次广义Γ-环的弱强分次B row n-M cC oy根.证明了任何一个分次广义Γ-环都有弱强分次B row n-M cC oy根和拟弱强分次B row n-M cC oy根,而且弱强分次B row n-M cC oy根小于等于拟弱强分次B row n-M c-C oy根.     

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.     

所有同态象其Jacobson根与单根相同的Γ-环

设M是Nobusawa意义下的Г-环,S.Kyuno定义了环M_2=其中R,L分别是M的右、左算子环.本文首先刻画了环M_2的本原理想与Ja-cobson根.其次引进了一类新的Г-环称为PM Г-环,建立了Г-环M、矩阵Г_(n,m)-环M_(m,n)、Г-环M的右(左)算子环R(L)、M-环Г及M_2的PM性质之间的关系.最后,给出了Г-环一般形式的Jacobson性质,Jacobson性质、Brown-McCoy性质以及PM性质为其特殊情况.     

SECOND-ORDER BOOLEAN ALGEBRAS

Opgedra aan Prof. Hennie Schutte by geleentheid van sy sestigste verjaarsdag.

Abstract

A Boolean algebra is the algebraic version of a field of sets. The complex algebra C(B) of a Boolean algebra B is defined over the power set of B; it is a field of sets with extra operations. The notion of a second-order Boolean algebra is intended to be the algebraic version of the complex algebra of a Boolean algebra. To this end a representation theorem is proved.     

分次环的分次根

给出了建立分次环根的一般方法．作为其应用，建立了分次环的分次Brown—McCoy根，并给出了Brown—McCoy半单分次环的结构定理．     

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.     

SHAPE AND INDUCED REPRESENTATIONS

Abstract

Let K: P → T be a fixed functor. A criterion is given for a functor M': T → V to be a (right) Kan extension along K of some functor M: P → V. The functors M having a given M' as Kan extension are, in general, classified by continuous functors (V ^P)^o → V. We introduce a notion of system of imprimitivity, generalizing that of Mackey; when the shape category of K is codense in the systems of imprimitivity classify the functors H having M' as Kan extension. As a special case one obtains Mackey's Imprimitivity Theorem for finite groups.     

PRODUCTS OF DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

To form products of differential expressions in the classical way it is necessary to place heavy differentiability assumptions on the coefficients. Here we consider symmetric (formally self-adjoint) expressions defined, not in the classical way, but in terms of quasi-derivatives. With this very general notion of symmetry we show that products such as M1M2MI of symmetric expressions M1, Hp can be formed vithout any smoothness assumptions on the coefficients and such products are symmetric expressions.     

THOUGHTS ON THE CANTOR-BERNSTEIN THEOREM

The usual proofs of the well-known set-theoretical theorem “Given one-one maps f: A → B and g:B → A, there exists a one-one onto map h:A → B” actually produce a map h:A → B contained in the relation f U g^?1. Considering Tarski's Fixpoint Theorem as the implicit basic ingredient of such proofs. We examine several classical proofs/starting with Dedekind (1887), and illuminate their common feature by means of the categorical notion of a natural fixpoint. We consider a categorical form (CBT) of the theorem (with h ? f Ug^?1) in a variety of contexts, obtaining some examples of categories where CBT holds and others where it fails. Among other results we prove for a topos E, (1) CBT holds if E is Boolean, and conversely if E has a natural number object; (2) The Axiom of Choice in E implies a dual version of CBTI and conversely if E has splitting supports and a natural number object.     

DISCONNECTEDNESS CLASSES

Abstract

Let X be an (E, M)-category for sinks. A notion of disconnectedness with respect to a closure operator C on X and to a class of X-monomorphisms N is introduced. This gives rise to the notion of N-disconnectedness class, a characterization of which is presented in a category with a terminal object. Some examples are provided.     

A note on Brown—McCoy radicals ofΓ-rings

In [1] we defined the Brown—McCoy radical,B(M), of a-ringM. In this note we show thatB is a special radical. The simplicial radical, defined by Kyuno [4] for-rings with left and right unities, is extended to arbitrary-rings. The simplicial radicalS is shown to be a generalization of the Brown—McCoy radical of a ring. In general,B(M) S(M). This work is supported financially by the Technikon Natal.     

PRECOMPACTNESS IN FUZZY UNIFORM SPACES

Abstract

The notion of a precompact fuzzy set in a fuzzy uniform space is defined and it is shown that this is a good extension of the standard notion. A theory of precompact fuzzy sets is developed using the previously defined notion of a Cauchy prefilter in a fuzzy uniform space and this theory generalises standard theory.     

J-半单亚直既约环类确定的上限

Ｆ．Ａ．Ｓｚａｓｚ在［１］中提出公开问题５５：设Ｋ是Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类,研究类Ｋ确定的上根．本文对此进行了研究,证明了Ｊａｃｏｂｓｏｎ根为零的全体亚直既约环类Ｋ确定的上根Ｒ是特殊根,它介于Ｊａｃｏｂｓｏｎ根与Ｂｒｏｗｎ－ＭｃＣｏｙ根之间．并给出任意结合环Ａ为Ｒ－根环的充要条件．     

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.     

A THEOREM OF BÄCKLUND REVISITED

Abstract

Bäcklund's theorem states that the most general contact transformation is an extended point transformation whenever both the number of independent variables and the number of dependent variables exceed one. A partial circumvention of Bäcklund's theorem is obtained by assigning each dependent variable its own distinct manifold of independent variables. This gives rise to extended symplectic product structures. sequences of extended Hamiltonians, and Lie groups of regular maps that satisfy systems of extended Hamilton-Jacobi equations provided the initial data is determined by a regular map. These ideas are applied to the study of systems of nonlinear second order partial differential equations. Lie groups of solutions are shown to be obtained by solving systems of extended Hamilton-Jacobi equations provided the initial data defines a solution.