讲座内容：

A classical open problem in ring theory is to characterize the integral domains R such that every singular matrix over R is a product of idempotent matrices. The importance of this problem is underlined by the inter-connections with other big unsolved questions: Classify integral domains whose general linear groups are generated by the elementary matrices and those verifying “weaker” versions of the Euclidean algorithm. In fact, over a Bézout domain, every singular matrix can be written as a product of  idempotent factors if and only if every invertible matrix can be written as a product of elementary matrices . Moreover, this happens if and only if the domain admits a weak algorithm.
In our talk, we will give an overview of classical results and recent developments on the factorization of matrices into idempotents. In particular, we will consider products of idempotent matrices over special classes of non-Euclidean principal ideal domains and over integral domains that are not Bézout.