Jaffard ring

In commutative algebra, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960.^[1]

Formally, a Jaffard ring is a ring R such that the polynomial ring

${\displaystyle \dim R[T_{1},\ldots ,T_{n}]=n+\dim R,\,}$

where "dim" denotes Krull dimension. A Jaffard ring that is also an integral domain is called a Jaffard domain.

The Jaffard property is satisfied by any Noetherian ring R, and examples of non-Noetherian rings might appear to be quite difficult to find, however they do arise naturally. For example, the ring of (all) algebraic integers, or more generally, any Prüfer domain.^[2] Another example is obtained by "pinching" formal power series at the origin along a subfield of infinite extension degree, such as the subring of ${\displaystyle {\overline {\mathbf {Q} }}[[T]]}$ consisting of those formal power series whose constant term is rational.^[3]