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. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. Then in general n ≥ dim A, and A is defined to be regular if n = dim A.
The appellation regular is justified by the geometric meaning. A point x on a algebraic variety X is nonsingular if and only if the local ring of germs at x is regular. Regular local rings are not related to von Neumann regular rings.
Read more about Regular Local Ring: Characterizations, Examples, Basic Properties, Origin of Basic Notions
Famous quotes containing the words regular, local and/or ring:
“They were regular in being gay, they learned little things that are things in being gay, they learned many little things that are things in being gay, they were gay every day, they were regular, they were gay, they were gay the same length of time every day, they were gay, they were quite regularly gay.”
—Gertrude Stein (1874–1946)
“In everyone’s youthful dreams, philosophy is still vaguely but inseparably, and with singular truth, associated with the East, nor do after years discover its local habitation in the Western world. In comparison with the philosophers of the East, we may say that modern Europe has yet given birth to none.”
—Henry David Thoreau (1817–1862)
“The life of man is a self-evolving circle, which, from a ring imperceptibly small, rushes on all sides outwards to new and larger circles, and that without end.”
—Ralph Waldo Emerson (1803–1882)