Mathematical Sciences: Noncommutative Rings

这个项目涉及非对易环理论中的一系列问题。主要的研究工作是：(I)同态映象生成的所有除法代数的构造；(Ii)证明了对于泛代数意义下的每两个簇U和V，U到V的可表示函子范畴有一个初始对象；(Iii)拓扑空间上集合的所有余层的刻画；(4)“Stone空间上的每个群结构都是有限群的逆极限”和“线性紧拓扑向量空间上的每个结合代数结构都是有限维代数的逆极限”等结果的一般推广；(V)关于除法代数的一种奇特构造的结果，它包括选择无限维向量空间V的子空间的一个非主极大滤子F，用核属于F的自同态环的理想构成V的自同态环的因子环。这是环论的一般领域。环是定义了加法和乘法的代数对象。虽然加法运算满足交换律，但乘法运算不是必须的。乘法不可交换的环的一个例子是整数上的n×n矩阵的集合。非对易环的研究已经成为代数的重要组成部分，因为它对数学和物理的其他分支的意义越来越大。

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.