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.Template:Sfnp

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

\dim R [T_{1}, \dots, 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.Template:Sfnp Another example is obtained by "pinching" formal power series at the origin along a subfield of infinite extension degree, such as the subring of $\overline{𝐐} [[T]]$ consisting of those formal power series whose constant term is rational.Template:Sfnp