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The notion of a convolution type is introduced. Imposing such a type on a ring gives the corresponding convolution ring. Under this umbrella, a wide variety of ring constructions can be covered, including polynomials, matrices, incidence algebras, necklace rings, group rings and quaternion rings. Here the influence of the convolution type on the corresponding convolution ring is investigated, in particular on the existence of homomorphisms and ideals.

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It was previously shown that every special radical class ~ of rings induces a special radical class p~ of r-rings. Amongst the special radical classes of nearrings, there are some, called the c~-special radical classes, which induce special radical classes of r-near-rings by the same procedure as used in the ring case. The a-special radicals of near-rings possess very strong hereditary properties. In particular, this leads to some new results for the equiprime and 273 radicals. O. IntroductionA most sought after and useful property of a radical ~ in an algebraic variety is the following: If I is an ideal of N E I), then 7~(I) = I N 7~(N). All the wellknown radicals in the variety of associative rings enjoy this property. However, this is not true of many of their generalizations to the variety of 0-symmetric near-rings. Nevertheless, it is well known that the Jacobson type radicals Z2 and 773, and the Brown-McCoy radical exhibit this property. Recently, it has been shown ([10, [11]) that several other radicals, based on the equiprime near-rings, also do so. In the present paper, we will show that many of these radicals satisfy a much stronger conditions, namely that 7~(I) = 1 N ~(N) for any ~,-subnear-ring (and hence any invariant subgroup) 1 of N. This leads us to introduce the concept of ~r-special radicals for 0-symmetric near-rings.In [3] it was shown that every special radical class 7~ of rings induces a special radical class pT~ of F-rings, as follows: Let A/[ be a special class of rings which determines 7~ as an upper radical. Let A74 be the set of all those F-rings M whose left operator ring L is in A//, and such that zrM = 0 implies z = 0, for all x E M. Then .~1 is a special class and pT~ is defined to be the upper radical determined by the class A/[. pT~ is uniquely determined in the sense that if )k41 and A/t2 are two special classes of rings which determine the same upper radical 7~, then the Mathematics subject classification numbers, 1991. Primary 16Y30, 16Y99.