Abstract:
The subject matter of this thesis concerns the study of pairs of classes of (zero-symmetric) right near-rings as well as pairs of ideal mappings with various radical theoretical properties associated with them. Relationships are expressed in terms of homomorphic images of near-rings belonging to the components of the class pair under consideration. We establish a framework for the study of these relationships by defining a class pair (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>). where <i>M</i><sub>1</sub> and <i>M</i><sub>2</sub> are non-empty classes of zero-symmetric near-rings. In addition to this, we define a radical pair (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) where <i>p</i><sub>1</sub> and <i>p</i><sub>2</sub> are pre- radical mappings and investigate links between (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) and related class pairs (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>). The concept of a radical pair (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) was first introduced by Snider in [50] for rings. Here <i>p</i><sub>1</sub> and <i>p</i><sub>2</sub> were required to be hereditary KA-radical mappings defined on the class of all rings. He showed that, in this case. (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) is a KA-radical class. Later Divinsky and Sulinski [16] continued the study of the class (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>), which they referred to as a radical pair. They showed that for specific choices of <i>p</i><sub>1</sub> and <i>p</i><sub>2</sub>, the class (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) describes the class of Jacobson rings (cf. [54]). In [49]. G. Shin considered the coincidence, of rings for which every prime ideal is completely prime and the rings in which the set of nilpotent elements coincides with the prime radical. This relationship led to the study in [7] where Birkenmeier, Heatherly and Lee formulated the 2-primal condition in the context of near-rings. An ideal <i>I</i> of a near-ring <i>R</i>. is a 2-primal ideal of <i>R</i> if β<sub>o</sub>(<i>R</i>/<i>I</i>) = <i>N</i>(<i>R</i>/<i>I</i>) where β<sub>o</sub>) is the 0-prime radical mapping for near-rings and <i>N</i>(<i>R</i>/<i>I</i>) is the set of all nilpotent elements of <i>R</i>/<i>I</i>. If the zero ideal of <i>R</i> is a 2-primal ideal of <i>R</i>, then <i>R</i> is a 2-primal near-ring. They obtained the following characterization of a near-ring <i>R</i> in which every ideal is a 2-primal ideal: Every ideal of <i>R</i> is a 2-primal if and only if every 0-prime ideal of <i>R</i> is completely prime. A general study of class pairs (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>) where <i>M</i><sub>1</sub> and <i>M</i><sub>2</sub> are non- empty sub-classes of some universal class of (associative) rings has recently been undertaken by Birkenmeier. Groenewald and Olivier in [4]. Their approach had the following point of departure. The class (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>) is homomorphically closed. With some added conditions on <i>M</i><sub>1</sub> or <i>M</i><sub>2</sub>. (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>) may exhibit some other radical theoretical properties. In Chapter 2 we extend this investigation to near-rings and give, amongst other things, sufficient conditions for (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>) to be a KA-radical class. In [47] it was noted by Puczylowski that for hereditary KA-radicals <i>p</i><sub>1</sub> and <i>p</i><sub>2</sub>, the radical pair (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) coincides with class pair (<i>Sp</i><sub>2</sub> : <i>Sp</i><sub>1</sub>) where <i>Sp</i><sub>2</sub>(<i>Sp</i><sub>1</sub>) denotes the semi-simple class of <i>p</i><sub>2</sub>(<i>p</i><sub>1</sub>). This immediately warrants a search for milder conditions under which this equality will hold. In the sequel, sufficient conditions are given for (<i>Sp</i><sub>2</sub>: <i>Sp</i><sub>1</sub>) = {<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) to hold where <i>Sp</i><sub>2</sub> and <i>Sp</i><sub>1</sub> are the semisimple classes of <i>p</i><sub>2</sub> and <i>p</i><sub>1</sub> respectively. Relationships between the class (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) and the class (<i>Rp</i><sub>1</sub> : <i>Rp</i><sub>2</sub>) are also described where <i>Rp</i><sub>2</sub> and <i>Rp</i><sub>1</sub> are the radical classes associated with <i>p</i><sub>2</sub> and <i>p</i><sub>1</sub> respectively. In this Chapter we require the components of (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>) to be classes of zero-symmetric near-rings and the components of (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) to be (more often than not) preradical mappings instead of KA-radical mappings. This is a more general setting than that of Birkenmeier, Groenewald and Olivier in [4]. Let us point out that even if <i>p</i><sub>1</sub> and <i>p</i><sub>2</sub> are KA-radical mappings, in this setting, the radical pair (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) will not always be a KA-radical class. However there is a guarantee that if <i>p</i><sub>1</sub> is a complete H-radical mapping and <i>p</i><sub>2</sub> an ideal-hereditary pre-radical mapping, then (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) will be a KA-radical class. This is a warning that many of the results obtained for the class and radical pairs of near-rings are not predictable analogues of the results obtained for class and radical pairs of associative rings. Let us recall that there are various types of primeness in near-rings, e.g. 0-primeness. 1-primeness etc. Similarly, there are various types of primitive ideals, e.g. 0-primitive. 1-primitive etc. The study of class pairs throws some light on the coincidence of prime ideals of one type with prime ideals of another type as well as the coincidence of the various types of prime ideals with the various types of primitive ideals. Similarly, coincidence of primitive ideals of one type with primitive ideals of another type is also studied in the class pair setting. This gives information about the radical properties of near-rings in which these types of ideals coincide. For example, let <i>P</i><sub>0</sub> be the class of 0-prime near-rings and <i>P</i><sub>c</sub> the class of completely prime near- rings. Then the near-rings in which every ideal is 2-primal can be expressed as the class pair (<i>P</i><sub>c</sub> : <i>P</i><sub>0</sub>). In general, we show that if <i>p</i><sub>2</sub> is an r-hereditary preradical mapping and <i>p</i><sub>1</sub> is the <i>H</i>-radical associated with <i>M</i><sub>1</sub> with both <i>Sp</i><sub>1</sub>and <i>Sp</i><sub>2</sub> hereditary, then (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) is a KA-radical class if either <i>M</i><sub>1</sub> (or <i>Sp</i><sub>1</sub>) is essentially closed or <i>M</i><sub>1</sub> (or <i>Sp</i><sub>1</sub>) is homomorphically closed. This is one of the main theorems of this chapter. This result is more general than the corresponding one in [4] which turns out to be a corollary of this result. We also show that conditions under which a class pair (<i>M</i><sub>1</sub> : <i>M</i><sub>2</sub>) forms a KA-radical class are similar to those for rings. Note that many of the results that are presented in this chapter will appear in [19]. Results from this chapter are also applied to solve an open question: Is the class of all 3-semiprime near-rings the semisimple class of the 3-prime radical? Or equivalently: Is every 3-semiprime ideal the intersection of 3-prime ideals? This question that was posed in [12]. [39] and [25] is answered in the negative. Towards the end of Chapter 2 we use class pairs to construct new s-radical mappings. The properties of these radicals are investigated and discussed in the special setting of Andrunakievic near-rings. Let <i>K</i> <| <i>I</i> <| <i>R</i> where <i>R</i> is a near-ring. It is well known that <i>K</i> is not necessarily an ideal of <i>R</i>. In the ring case. Andrunakieviclr's lemma asserts that (<<i>K</i> >><sub><i>R</i></sub>)<sup>3</sup> ⊆ <i>K</i>. It. is also well known that this useful ring theoretic result regrettably does not carry over to near-rings. However, there are near-rings for which there exists a natural number <i>n</i>. depending on each ideal <i>I</i> of <i>R</i>. such that. <i>K</i> <| <i>I</i> <| <i>R</i> implies (< <i>K</i>><sub><i>R</i></sub>)<sup>3</sup> ⊆ <i>K</i>. Such ideals are called Andrunakievic ideals (<i>A</i>-ideals for short). If all the ideals of a near-ring <i>R</i>, are <i>A</i>-ideals, then <i>R</i> is called an Andrunakievic near-ring (<i>A</i>-near-ring for short). In this setting we require that the radical pair (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) be closed under the taking of sums of ideals. This opens a number of new possibilities. For each pair (<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>) we construct an idempotent preradical β<sub>(<i>p</i><sub>1</sub> : <i>p</i><sub>2</sub>)</sub>. This leads to a generalization of the results obtained in [4]. In Chapter 3 the focus is placed on the various primeness conditions for near-rings and properties that will raise primeness from one level to the next. The <i>i</i>-prime (<i>i</i>-semiprime) condition, <i>i</i> ⊆ (0,1, 2, 3, <i>e</i>, <i>c</i>), for ideals of near- rings is studied and. in some cases, linked via the concept of <i>i</i>-reflexivity of an ideal or (<i>i</i>,<i>j</i>)-ideal of <i>R</i>, <i>i</i> ⊆ {0,1, 2,3, <i>c</i>}. The relationships between class pairs determined by <i>i</i>-prime (<i>i</i>-semiprime) near-rings and class pairs determined by fully (<i>i</i>,<i>j</i>)-near-rings are also given. As a result, new ways of describing fully <i>i</i>-primal (<i>i</i>-semiprimal) near-rings are possible. As a specific application, the class of 2-primal (2-semiprimal) near-rings, that was studied by Birkenmeier et. al. in [6]. is characterized in a different way. The light thrown by class pairs on zero-symmetric near-rings goes further. It creates, in Chapter 4, an opportunity to explore the conditions under which a class pair or a radical pair will form a special radical class. In addition, the opportunity to introduce Jacobson near-rings as a generalization of Jacobson rings is seized in this chapter. This notion is motivated by a paper of Watters [55] that highlighted the importance of the class of Jacobson rings. We also exploit the notion of matrix near-rings as introduced by Meldrum and van der Walt [41] to study class pairs. We succeed in showing that most classes of Jacobson near-rings are matrix extensible and that some are KA-radical classes and even special radical classes. This shows that, in some cases, the property of being Jacobson is preserved in the transition from the near-ring to its corresponding matrix near-ring and conversely. Moreover, this notion gives a characterization of the class pairs (<i>S<sub>Jv</sub</i> : <i>P<sub>i</sub></i>) where (<i>S<sub>Jv</sub</i> is the semisimple class of the <i>J<sub>v</sub</i>-radical and <i>P<sub>i</sub></i> is the class of the <i>i</i>-prime near-rings for <i>i</i> ⊆ {0, 2, 3, <i>e</i>, <i>c</i>} and <i>v</i> ⊆ {0,1, 2, 3}. In this final chapter it is shown that, by appropriately choosing the prime radical for near-rings, the 2-primal class of rings extended to near-rings can either be the radical class of some H-radical, a KA-radical class or a special radical class.