We consider universal classes of multioperator groups, and give a sufficient condition for a subclass defined by algebraic elementwise rules to be a radical class.Consider a universal class 14 of multioperator groups. We obtain a sufficient condition for algebraic element-wise definitions of classes of algebras in U to give rise to radical classes. Some work of this sort has been done by Gardner [1] and Wiegandt [5]. As our work is largely motivated by rings, we refer to normal subobjects as ideals throughout. We direct the reader to [2] for terminology and background.DEFINITION 1: Let U be a universal class in a variety V of multioperator groups, and let F be a set of elements in the free algebra with countable generators {xi, X2, • • . } in V. Let TZF be the class in U defined as follows: R is in TZF providing that for every r £ R there exist f(xi,X2,-.. ,x n ) £ F and r 2 ,r3,... ,r n in R, such that f(r,r 2 ,r 3 ,. ..,r n )-0.For any algebra R, define TZF'(R) = {r £ R | there exists / £ F, and r 2 , . . . , r n 6 R with /(r,r 2 > ... ,r n ) = 0}. It is obvious from these definitions that R is in TZF if and only if TZF'(R) = R. LEMMA 2 . For all F, TIF is homomorphically closed.PROOF: Suppose R £ TZF, I

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We consider a generalisation of the Kurosh-Amitsur radical theory for rings (and more generally multi-operator groups) which applies to 0-regular varieties in which all operations preserve 0. We obtain results for subvarieties, quasivarieties and element-wise equationally defined classes. A number of examples of radical and semisimple classes in particular varieties are given, including hoops, loops and similar structures. In the first section, we introduce 0-normal varieties (0-regular varieties in which all operations preserve 0), and show that a key isomorphism theorem holds in a 0-normal variety if it is subtractive, a property more general than congruence permutability. We then define our notion of a radical class in the second section. A number of basic results and characterisations of radical and semisimple classes are then obtained, largely based on the more general categorical framework of L. Márki, R.Mlitz and R. Wiegandt as in [13]. We consider the subtractive case separately. In the third section, we obtain results concerning subvarieties and quasivarieties based on the results of the previous section, and also generalise to subtractive varieties some results for multi-operator group radicals defined by simple equational rules. Several examples of radical and semisimple classes are given for a range of fairly natural 0-normal varieties of algebras, most of which are subtractive. 2 GeneralitiesWe are interested in 0-regular varieties in which {0} is always a subalgebra; this includes of course all varieties of multi-operator groups, but also loops, hoops and various other fairly natural examples, which we return to later. 0-regularity of a variety can be described in terms of a so-called

We generalise some existing results on strict radical classes, and introduce some new notions of hereditariness of radical classes. This is achieved by considering the classes of rings for which a radical class is hereditary and/or strict (in the general sense).