Regular local ring

In mathematics, and more specifically in commutative algebra, a regular local ring is a local Noetherian ring that has the property that the minimum number of generators of its maximal ideal (also called ideal ideal) is exactly the same as its Krull dimension . The minimum number of generators of the maximal ideal is always bounded lower by the Krull dimension. Formally, if A is a local ring with maximal ideal m, and suppose that m is generated by a1, ..., an, then n ≥ dim A, and A is regular if and only if n = dim A. >

The denomination of regular is justified by its geometric meaning: a point of an algebraic variety is non-singular if and only if the associated local ring is regular.

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