Arf ring
Encyclopedia
In mathematics
, an Arf ring was defined by to be a 1-dimensional
commutative
semi-local
Macaulay ring
satisfying some extra conditions studied by .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an Arf ring was defined by to be a 1-dimensional
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
commutative
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
semi-local
Semi-local ring
In mathematics, a semi-local ring is a ring for which R/J is a semisimple ring, where J is the Jacobson radical of R. The above definition implies R has a finite number of maximal right ideals...
Macaulay ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
satisfying some extra conditions studied by .