Regular local ring

In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension.^[1] In symbols, let ${\displaystyle A}$ be any Noetherian local ring with unique maximal ideal ${\displaystyle {\mathfrak {m}}}$, and suppose ${\displaystyle a_{1},\cdots ,a_{n}}$ is a minimal set of generators of ${\displaystyle {\mathfrak {m}}}$. Then Krull's principal ideal theorem implies that ${\displaystyle n\geq \dim A}$, and ${\displaystyle A}$ is regular whenever ${\displaystyle n=\dim A}$.

The concept is motivated by its geometric meaning. A point ${\displaystyle x}$ on an algebraic variety ${\displaystyle X}$ is nonsingular (a smooth point) if and only if the local ring ${\displaystyle {\mathcal {O}}_{X,x}}$ of germs at ${\displaystyle x}$ is regular. (See also: regular scheme.) Regular local rings are not related to von Neumann regular rings.^[a]

For Noetherian local rings, there is the following chain of inclusions:

Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings

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