chapter  17
10 Pages

Pru¨fer Conditions in Rings with Zero-Divisors

BySarah Glaz

In his article “Untersuchungen u¨ber die Teilbarkeitseigenschaften in Ko¨rpernn,” J. Reine Angew. Math. 168, 1 - 36, 1932 [21], Heinz Pru¨fer introduced a new class of integral domains, namely those domains R in which all finitely generated ideals are invertible. He also proved that to verify this condition, it suffices to check that it holds for all two-generated ideals of R. This was the modest beginning of the notion of a Pru¨fer domain, a notion which made, and continues to make, a significant impact on research in non-Noetherian commutative ring theory. Heinz Pru¨fer (1896-1934), in his short life, had no opportunity to see the rings named in his honor by Krull ([17], 1936). It is not an exaggeration to say that today there is no conference on a non-Noetherian ring theory topic where the notion of a Pru¨fer domain does not make an appearance.