# Professor Emeritus Laszlo Fuchs' 90th Birthday Celebration

### Luigi Salce (Universita di Padova, Italy)

"Products of Elementary and Idempotent Matrices over Integral Domains"

A ring R such that invertible matrices over R are products of elementary matrices is called (after Cohn) generalized Euclidean. Characterizations of generalized Euclidean commutative domains, extending those proved by Ruitenburg for Bezout domains, are illustrated. These results connect generalized Euclidean domains with those domains R such that singular matrices over R are products of idempotent matrices. This latter property is investigated, focusing on 2x2 matrices, which is not restrictive in the context of Bezout domains. The relationship with the existence of a weak Euclidean algorithm, a notion introduced by O'Meara for Dedekind domains, is also described.

### Kulumani Rangaswamy (UC-Colorado Springs, USA)

"A Commutative Algebra Retrospective of the Work of Laszlo Fuchs"

Laszlo Fuchs started off his research career as a commutative ring theorist. During the first decade of his career, his primary focus was on the ideal theory of rings, particularly, the existence and uniqueness of representations of an ideal in a commutative ring as the intersection of many new types of ideals. These results generalized Emmy Noether's theorems in several ways. This earned him) just seven years after finishing his thesis) the Kossuth Prize. the coveted highest national award for scientists in Hungary. After impressive and far-reaching detours into abelian groups, ordered algebraic systems and module theory, Laszlo has turned his attention during the last decade back to the ideal theory of commutative rings, thus completing a full circle of his amazing journey through several important areas of research. This talk will describe some of the initial work on the ideal theory of commutative rings by Laszlo, and point out its connection to his more recent work on the same topic.

### Bruce Olberding (New Mexico State University, USA)

"A Look Ahead to New Problems and Directions Inspired by the Work of Laszlo Fuchs"

Much of the early work of Laszlo Fuchs in commutative ring theory focused on ideal decompositions, generalizations of primary ideals, and the lattice of ideals of a commutative ring. His work introduced a number of important ideas that remain influential in the field today. In this talk, we look at how some of these ideas are brought to bear in current research in decomposition theory, real algebraic geometry and the theory of associated primes.