Mathematical Sciences: Noncommutative Rings

数学科学：非交换环

• 批准号: 9303379
• 负责人: George Bergman
• 金额: $ 16.26万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9303379&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Noncommutative; Rings

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.

本项目涉及非交换环理论中的广泛问题。首席研究员将致力于(i)由同态象生成的所有除法代数的构造；（ii）证明对于每两个泛代数意义上的变量U和V， U到V的可表征函子的范畴有一个初始对象；（iii）拓扑空间上集合的所有共捆的表征；（iv）“Stone空间上的每一个群结构都是有限群的逆极限”和“线性紧致拓扑向量空间上的每一个关联代数结构都是有限维代数的逆极限”等结果的一般推广；(v)关于除法代数的一个奇特构造的结果，该构造包括在无限维向量空间v的子空间中选择一个非主极大滤波器F；由核属于F的V自同态的理想构成遵从F的V自同态环的因子环，本研究属于环理论的一般领域。环是一个代数对象，在它上面定义了加法和乘法。虽然加法运算满足交换律，但乘法运算不需要满足交换律。一个乘法不可交换的环的例子是整数上的nxn个矩阵的集合。非交换环的研究由于其对其他数学和物理分支的重要意义而成为代数的重要组成部分。