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The cubic plane curve (red) defined by the equation y 2 = x 2 (x + 1) is singular at the origin, i.e., the ring k[x, y] / y 2 − x 2 (x + 1), is not a regular ring. The tangent cone (blue) is a union of two lines, which also reflects the singularity. The k-vector space m/m 2 is an algebraic incarnation of the cotangent space.
A ring is a set R equipped with two binary operations [a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms: [1] [2] [3] R is an abelian group under addition, meaning that: (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative). a + b = b + a for all a, b in R (that ...
As an example, the nilradical of a ring, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n × n matrices over a division ring never forms an ideal, irrespective of the division ring chosen.
A commutative ring (not necessarily a domain) with unity satisfying this condition is called a containment-division ring (CDR). [2] Thus a Dedekind domain is a domain that either is a field, or satisfies any one, and hence all five, of (DD1) through (DD5). Which of these conditions one takes as the definition is therefore merely a matter of taste.
In algebra, a unit or invertible element [a] of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that = =, where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u.
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
A commutative algebra is an associative algebra for which the multiplication is commutative, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification.
This follows, for example, by Miracle Flatness: R is finite over the polynomial ring A = K[y], with degree 1 over points of the affine line Spec A with y ≠ 0, but with degree 2 over the point y = 0 (because the K-vector space K[x]/(x 2) has dimension 2). If K is a field, then the ring K[x,y,z]/(xy,xz) (the coordinate ring of the union of a ...