Strongly hereditary radicals

Abstract

A radical class (P of some universal class SW of not necessarily associative rings is hereditary if RCG implies IC'( for all ideals I of R, and will be called strongly hereditary (terminology suggested by M. Slater) if for all R CW we have G(I) = InvP(R) for all ideals I of R. (Note that in [2 ] it is the classes with this last property which are called hereditary.) It is well known that these properties are equivalent if SW is any universal class of associative or alternative rings [1, Lemma 1, p. 595 together with Corollary 1, p. 597 for associative rings or Corollary 1, p. 602 for alternative rings]. It is also clear that every strongly hereditary class is hereditary, but the converse is not true in general. This was shown by E. P. Armendariz in [2] using either of the rings of [3, p. 1116]. Note that by [1, Lemma 2, p. 595] hereditary radical classes are strongly hereditary if their semisimple classes are also hereditary. Since the hereditary and strongly hereditary properties are in general distinct, E. P. Armendariz has raised the question (private communication) of the existence of a strongly hereditary radical construction analogous to the lower radical. The key to such a construction is provided by the following alternative criterion for a strongly hereditary radical: