Grant: Discovery Grants Program - Individual

My research is in nonassociative ring theory and loop theory. A "ring" is a set of elements, like the integers, which you can add and multiply. Unlike the integers, however, I don't always assume that multiplication is associative: thus (ab)c=a(bc) need not always be the case. A long-standing project of mine has to been to find all small rings of various types. An important class of rings which are not (necessarily) associative is the class of "alternative" rings, these being rings in which (ab)c=a(bc) is not generally true though it is true if two of the three symbols are the same. Some years ago, a PhD student and I found all the alternative "algebras" of low prime power orders, work I want to continue and for other kinds of rings as well. A "loop" has just one operation. Again, I don't assume associativity, although I have become interested in certain loops that are "nearly associative" in the sense that all their proper subloops are associative. Together with a colleague in Philadelphia, some preliminary results have been obtained in special cases. There is an object called a "loop ring" whose elements look like polynomials, a+bx+cx^2+ ..., but where the powers of x are replaced by elements from a loop. When multiplication in the loop is associative, the resulting "group ring" has been studied a lot. Much of my research involves attempts to push known results from group rings to more general loop rings and to find features unique to this situation. The continuation of this work is part of this proposal as it is the search for loops whose loop rings satisfy various identities. My work is well known and has brought Canada and the name of Memorial University to the attention of people in Hungary, the Czech Republic, and Brazil, for example. As well, quite a bit of my research is accessible even to undergraduate students whom I hope to continue to attract to mathematical research.