Rings are a generalization of integers and polynomials, and they were first formalized by Dedekind in the 1870s. They have a variety of uses, including in number theory and algebraic geometry.
A ring is a set equipped with two operations (usually referred to as addition and multiplication) that satisfy certain properties: there are additive and multiplicative identities, addition is commutative, and the operations are associative and distributive. The simplest prototypical example is the ring of integers with the operations of addition and multiplication.
The algebra R displaystyle mathbb R -algebra of functions is the ring from X to R with the natural operations pointwise addition and multiplication of functions. It has several interesting subrings constructed by restricting to functions with properties that are preserved under addition and multiplication, such as continuous, differentiable, and polynomial functions.
There are also rings that satisfy other requirements. For instance, if a ring is commutative, then there must be an element 1 in R such that for all elements a in R, a1 = a. These rings are called commutative rings, and are fundamental objects in number theory, algebraic geometry, and many other areas of mathematics.
If every nonzero element in a commutative ring with unity has a multiplicative inverse, the ring is called a field. These fields are a fundamental object in number theory and algebraic geometry, and they have applications in a wide range of other mathematical fields, such as combinatorics and analysis.
In addition, if the ring is of a special type, such as a division ring or a skew field, it can be shown to be a commutative monoid. This is done by imposing the axiom that 0 a = a 0 for all a in R.
The ring can also be obtained by weakening the assumption that R is an abelian group to the assumption that it is a commutative monoid, and adding the axiom that a is an additive identity (called 0). This is known as a semiring or rig.
In algebraic topology, a ring spectrum is the morphism m : X X – X displaystyle mu :Xwedge X to X that gives rise to a sphere spectrum S with a unit map S – X from S, such that all ring axiom diagrams commute up to homotopy. These ring axiom diagrams are a useful tool for investigating algebraic structures, such as enveloping quotients and quotient groups.