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:
““I couldn’t afford to learn it,” said the Mock Turtle with a sigh. “I only took the regular course.”
“What was that?” inquired Alice.
“Reeling and Writhing, of course, to begin with,” the Mock Turtle replied; “and then the different branches of Arithmetic—Ambition, Distraction, Uglification, and Derision.”
“I never heard of ‘Uglification,’” Alice ventured to say.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)
“America is the world’s living myth. There’s no sense of wrong when you kill an American or blame America for some local disaster. This is our function, to be character types, to embody recurring themes that people can use to comfort themselves, justify themselves and so on. We’re here to accommodate. Whatever people need, we provide. A myth is a useful thing.”
—Don Delillo (b. 1926)
“Tell me where is fancy bred,
Or in the heart or in the head?
How begot, how nourished?
Reply, reply.
It is engendered in the eyes,
With gazing fed, and fancy dies
In the cradle where it lies.
Let us all ring fancy’s knell.
I’ll begin it. Ding, dong, bell.”
—William Shakespeare (1564–1616)