What Are Rings?

Rings are algebraic structures that generalize fields. In particular, they are equipped with two binary operations displaystyle a + b = a displaystyle a+b=b+a that satisfy properties analogous to those of the ordinary operations of addition and multiplication of integers.

Ring elements may be numbers such as integers or complex numbers, but they also include non-numerical objects such as polynomials, square matrices, functions and power series. In addition, the term is used in other fields of mathematics such as commutative algebra and algebraic topology, to describe the behavior of ring axiom diagrams.

In commutative algebra, rings are treated as monoid objects in good category of spectra, and their axiom diagrams commute up to homotopy. In the theory of abelian groups, a ring is a group under addition, closed, associative, distributive and bears a curse.

The simplest ring is the set of even integers with the usual + and. Other ring elements include polynomials and in one and two variables, as well as square real matrices.

Several mathematical theories are associated with a specific ring, and the corresponding ring is named after its investigators. Unfortunately, this practice often leads to names that do not provide useful insight into the relevant properties of the associated rings.

A commutative ring is a ring which has a commutative multiplicative identity for every a in R. This property is called a ring homomorphism, and it is the main subject of commutative algebra.

To construct a ring, consider an abelian group A which is associated to the set of all morphisms m of R. In particular, m(r x) is a morphism of A by right (or left) multiplication by r.

In addition, m(r x) is an endomorphism of A. Moreover, m(r x) is factor through m(r x). This is an axiom which is usually applied to all a in R, since it allows the morphisms of A to be factored.

There are also commutative semirings, which are a type of commutative monoid. These rings have a commutative multiplicative inverse and no divisors of zero.

An important branch of ring theory is the study of commutative rings, which are a generalization of Dedekind domains that occur in number theory and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Using these commutative rings, a variety of other mathematical theories have been developed such as spectral theory and commutative ring spectrum.

Among the most prominent contributions of ring theory are the concepts of axiomatic symmetry, which allows a commutative ring to be represented by a symmetric spectral graph, and symmetric morphisms, which are a commutative ring’s symmetric substructure. These concepts are also used in many other fields of mathematics such as algebraic topology, and are derived from the ideas of axiomatic symmetry.

Besides a ring’s commutative status, it also has to have a unit element and no divisors of zero. For example, a ring of arithmetic arcs of finitely many polynomials is an arithmetic ring and contains only one unit element. Other arithmetic rings are cyclic, which contain only a single unit element. Various other types of arithmetic rings are also known, including infinite rings, composite rings and rings of complex numbers.