A.J. Van der Walt

Centralizer near-rings over free ring modules

We treat centralizer near-rings over ring modules in general, with particular emphasis on the case of free modules. Questions like the following are answered. When is the near-ring a nonring? When is the near-ring simple? What are its maximal and minimal left ideals? What is its subgroup structure? What is the radical? The cases where the ring concerned is a PID or a field are treated in some detail.

Centralizer near-rings determined by PID-modules, II

We answer an open problem in radical theory by showing that there exists a zero-symmetric simple near-ringN with identity such thatJ2(N)=N.

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.

PIECEWISE ENDOMORPHISMS OF RING MODULES

Abstract We study centralizer near-rings of ring modules which are rather special in two respects. Firstly, the elements of the near-rings are piecewise endomorphisms of the modules concerned. and secondly, the near-rings themselves are, in fact, rings.

Parody as a Means to Advance the Objectives of Copyright Law

Contemporary South African culture is riddled with parodies of every kind. Although a parody culture exists in practice, artists and other creators have a double-barrelled shotgun to face: they have to worry not only about infringing the dignity of the subject of their parody, but also infringing copyright if a creative work protected by copyright is their subject. The controversial painting The Spear by Brett Murray illustrates this conundrum, where the subject of the parody was not only President Jacob Zuma, but also the Soviet propagandist poster which originally featured Vladimir Lenin. As we shall show below, South African copyright law allows a work to be used for a number of purposes, including criticism and review (which is closely related to parody, as ridicule is often a form of criticism), but makes no concessions for blatant comedic expression as a defence to a copyright infringement claim. In the absence of a specific fair dealing exception for this purpose, the public is free to parody elements of South African culture, provided they refrain from using a substantial part of a work portraying the particular feature they wish to parody. If a user wanted to level criticism against a copyright work by means of parody, or use the characteristic elements of a work in a transformative and humorous way, he would find himself without protection. In this way the current copyright regime prevents many new works from coming to fruition, as users have to obtain permission from the author for the parody to be created.

Near-Rings and Partitioned Groups

The near-rings of mappings arising from partitioned groups have been studied by Maxson and van der Walt (1994). We axiomatize the situation giving rise to these near-rings, defining a near-ring associated with a family of partitioned groups. This generalizes the work of Maxson and van der Walt and there are many special cases which are of interest. We develop a general structure theory for these near-rings and we also look in more detail at some special cases.

Krull Dimension and Tame Near-Rings

A tame near-ring R is one which has a faithful module in which all R-modules are R-ideals. Such a near-ring is very ring-like. In this paper we extend some results on rings with Krull dimension to tame near-rings. Krull dimension is a generalization of the minimal condition. The main results are as follows (i) A tame right noetherian near-ring has a possibly transfinite power of its J 2 radical zero, (ii) The J 2 radical of a tame near-ring does not contain a non-zero idempotent ideal, (iii) The nil radical of a tame near-ring, is residually nilpotent.