 Informally, the elements of "m" can be thought of as functions which vanish at the point "p", whereas "m"2 contains the ones which vanish with order at least 2.For any Noetherian local ring "R", the inequality
holds true, reflecting the idea that the cotangent (or equivalently the tangent) space has at least the dimension of the space Spec "R".If equality holds true in this estimate, "R" is called a regular local ring.