Report title: On the Small Finitistic Dimension of Prufer Ring

Reporter: Professor Wang Fanggui, School of Mathematics, Sichuan Normal University

Reporting time: 14:00-15:00, September 15, 2020

Report location: Tencent Conference ID: 644 468 534

Or click the link to join the meeting directly:

https://meeting.tencent.com/s/EQqdIiWmci8P

School contact: Sun Xiaosong sunxs@jlu.edu.cn

Report summary:

In 1932, Prufer introduced a type of integral ring, that is, each non-zero finitely generated ideal is an integral ring of a reversible ideal. Equivalently, w.gl.dim(R)≤1. In 1936, Krull named it Prufer ring. Because Prufer ring has many good ring structure theories and homology properties, in 1970, Griffin established this concept on the general commutative ring, leading to the concept of Prufer ring. The commutative ring R is called Prufer ring, It means that each finitely generated regular ideal is a reversible ideal. In 2006, Bazzoni and Glaz collected the research results of Prufer ring, but these results were obtained from the perspective of multiplicative ideal theory. In 2014, Gahen-Fontana-Frisch- Glaz asked the following public questions:

Problem 1a: Let R be a Prufer ring. Is fPD(R) ≤1?

Problem 1b: Let R be a total ring of quotients. Is fPD(R)=0?

  We use a finite generation semi-regular ideal multiplication system to establish a Lucas module system similar to w-modules, then construct counterexamples, and give negative answers to the above two public questions.

Brief introduction of the speaker:

Wang Fanggui，professor in the School of Mathematics of Sichuan Normal University, doctoral supervisor, commentator of German "Mathematics Digest", a member of the China Democracy Promotion Association, and a representative of the 14th and 15th People's Congress of Chengdu，engaging in the research of exchange algebra, homology algebra, algebra K-theory, ring and module category theory, cryptography, presided over a number of projects such as the National Natural Science Foundation of China and the Doctoral Fund of the Ministry of Education, and published more than 100 academic papers. It is highly valued by domestic and foreign peer experts, and many of his research papers, especially those about w-operator, have been cited many times by domestic and foreign colleagues. He has won the third prize of Jiangsu Science and Technology Progress Award, the third prize of Science and Technology Progress Award of the State Education Commission, the second prize of Science and Technology Award of Chinese Universities, and the second prize of Sichuan Education Achievement Award.