From Krasner’s corpoid and Bourbaki-Krasner’s graded rings to Krasner-Vuković’s paragraded rings
Marc Krasner investigating valued fields and observing a connection between them and their valuation rings using equivalence of valuations came to the fundamental notion of a corpoid in a series C.R. of Academy of Sciences of Paris notes (1944-45), which represents the origin of the development forerunner of the general graded theory – a theory of homogroupoids, anneids and moduloids – more general than one given by Bourbaki (Algèbre, Chap II Paris, 1962), since neither the associativity, nor the commutativity nor the existence of a neutral element is assumed in the set of grades, where corpoid, as a special case of an anneid, is viewed
as a homogeneous part of a graded field. So, the abstract notion of corpoid led Krasner to a development of general graded structures (Anneaux gradués géneraux, 1980)
Since the category of graded structures (groups, rings, modules) has no the property of closure with respect to the direct product and the direct sum, it was for M. Krasner and myself a motivation to go further with generalizations and to introduce the notion of paragraded structures: groups, rings, modules, which appeared for the first time, late in 1980s, in a series Krasner- Vuković’s Proceedings Japan Academy notes and in our monograph Structures Paragraduée (groupes, anneaux, modules) (Queen’s Papers in pure and Applied Mathematics, Kingston, ONT. Canada). As we have already noted, these structures resolves the formentioned problem of closure. In this presentation we are particularly interested in the structure of a paragraded ring and their radicals, and we are going to present some results on the different types of paragraded radicals, introduced in my joint papers with my pupils, in the class of the paragraded rings. More precizely we will present prime and Jacobson radicals, discuss the general Kurosh-Amitsur theory of radicals of paragraded rings, characterise paragraded normal radicals and prove that all special paragraded radicals of paragraded rings can be described by the appropriate class of their paragraded modules and finally provide information about ongoing work regarding paragraded Brown-McCoy radicals, i.e. inspired by Halberstadt results about Jacobson’s radicals of graded rings, to introduce two versions of paragraded Brown-McCoy radicals: the Brown- McCoy radical and the large Brown-McCoy radical of a paragraded rings and to prove that the large Brown-McCoy radical of a paragraded ring coincides with the largest homogeneous ideal contained in the classical Brown-McCoy radical of that ring.