In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not or is not required to be commutative. e, operations combining any two elements of the ring to a third. They are called addition and multiplication and commonly denoted by + and ⋅, e. g. a + b, the identity elements for addition and multiplication are denoted 0 and 1, respectively. If the multiplication is commutative, i. e. a ⋅ b = b ⋅ a, in the remainder of this article, all rings will be commutative, unless explicitly stated otherwise. An important example, and in some sense crucial, is the ring of integers Z with the two operations of addition and multiplication, as the multiplication of integers is a commutative operation, this is a commutative ring. It is usually denoted Z as an abbreviation of the German word Zahlen, a field is a commutative ring where every non-zero element a is invertible, i. e. has a multiplicative inverse b such that a ⋅ b =1. Therefore, by definition, any field is a commutative ring, the rational, real and complex numbers form fields. An example is the set of matrices of divided differences with respect to a set of nodes. If R is a commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms the polynomial ring. The same holds true for several variables, if V is some topological space, for example a subset of some Rn, real- or complex-valued continuous functions on V form a commutative ring. The same is true for differentiable or holomorphic functions, when the two concepts are defined, such as for V a complex manifold, in contrast to fields, where every nonzero element is multiplicatively invertible, the theory of rings is more complicated. There are several notions to cope with that situation, first, an element a of ring R is called a unit if it possesses a multiplicative inverse. Another particular type of element is the zero divisors, i. e. a non-zero element a such that there exists an element b of the ring such that ab =0. If R possesses no zero divisors, it is called an integral domain since it resembles the integers in some ways. Many of the following notions also exist for not necessarily commutative rings, for example, all ideals in a commutative ring are automatically two-sided, which simplifies the situation considerably. Given any subset F = j ∈ J of R, the ideal generated by F is the smallest ideal that contains F. Equivalently, an ideal generated by one element is called a principal ideal. A ring all of whose ideals are principal is called a principal ideal ring, any ring has two ideals, namely the zero ideal and R, the whole ring
Spec (Z) contains a point for the zero ideal. The closure of this point is the entire space. The remaining points are the ones corresponding to ideals (p), where p is a prime number. These points are closed.
The twisted cubic (green) is a set-theoretic complete intersection, but not a complete intersection.