Pru¨fer Conditions in Rings with Zero-Divisors
In his article “Untersuchungen u¨ber die Teilbarkeitseigenschaften in Ko¨rpernn,” J. Reine Angew. Math. 168, 1 - 36, 1932 , 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 (, 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.