Mathematical Sciences: Noncommutative Rings

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This project is concerned with a wide spectrum of problems in noncommutative ring theory. The principal investigator will work on (i) a construction of all division algebras generated by homomorphic images; (ii) a proof that for every two varieties U and V in the sense of universal algebra, the category of representable functors of U into V has an initial object; (iii) a characterization of all co-sheaves of sets on a topological space; (iv) a common generalization of such results as "every group structure on a Stone space is an inverse limit of finite groups" and "every associative algebra structure on a linearly compact topological vector space is an inverse limit of finite-dimension algebras"; (v) results on a curious construction for division algebras, which consists of choosing a nonprincipal maximal filter F of subspaces of an infinite-dimensional vector space V, and forming the factor-ring of the ring of endomorphisms of V that repect F by the ideal of those endomorphisms whose kernel belongs to F. This research is in the general area of ring theory. A ring is an algebraic object having both an addition and a multiplication defined on it. Although the additive operation satisfies the commutative law, the multiplicative operation is not required to do so. An example of a ring for which multiplication is not commutative is the collection of nxn matrices over the integers. The study of noncommutative rings has become an important part of algebra because of its increasing significance to other branches of mathematics and physics.