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We prove that rings of weak dimension one are the rings with all three generated ideals syzygetic This leads to a characterization of these rings in terms of the Andr e Quillen homology Let I be an ideal of a commutative ring A There is a canonical morphism of graded A algebras S I R I from the symmetric algebra of I onto its Rees algebra The ideal I is said to be of linear type if is an isomorphism If S I I is an isomorphism I is said to be syzygetic In C Theorem Costa showed that a domain A is Pr ufer if and only if I is of linear type for every two generated ideal I of A and I is syzygetic for every three generated ideal I of A In this note we show that the preliminary hypothesis that A is a domain can be removed by changing the Pr ufer condition to the condition wd A weak dimension of A one or less Moreover the condition that every two generated ideal of A be of linear type is not necessary Concretely Theorem Let A be a commutative ring The following conditions are equivalent i wd A ii Every ideal of A is of linear type iii Every ideal of A is syzygetic iv Every three generated ideal of A is syzygetic Recall that wd A is the supremum of the at dimensions of all A modules Von Neu mann regular rings are those of weak dimension zero Semihereditary rings i e rings with all its nitely generated ideals projective have weak dimension one or less In fact A is a semihereditary ring if and only if wd A and A is coherent A semihereditary domain is called a Pr ufer ring For a domain A to be Pr ufer is equivalent to wd A see B Mathematics Subject Classi cation Primary F A Secondary D

Rings of weak dimension one and syzygetic ideals