In mathematics, rings are sets equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. In general, the elements of a ring are numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions and power series.
There are many interesting classes of rings. Some of the most common are commutative rings (i.e., if two elements are multiplied they must commute), associative rings (i.e., a and b can be multiplied in any order), and rings with unity (i.e., a1 = a for all a in the ring).
One of the most important branches of ring theory is commutative algebra. This is the area of research devoted to studying rings that are commutative under multiplication, which is a much more complicated topic than non-commutative rings.
A ring can be defined as a set of numbers that satisfy the following requirements:
The first requirement is that every element must commute with all other elements in the ring. In addition, the ring must have a zero element and negatives of each element.
Another important requirement is that the ring must be associative under addition. This requires that the ring has two distributive laws of addition: a(b + c) = ab and a(c + b)c.
Moreover, if we restrict the ring to functions with properties that are preserved under addition and multiplication, then we get interesting subrings of the ring. These include continuous functions, differentiable functions, polynomial functions and so on.
These ring-like groups are called fields, and some of them may be of interest for practical applications. In particular, they are useful for constructing arithmetic circuits.
Other important properties of fields are that they can be grouped together, that they can have a unit element, and that their divisors cannot be zero. This leads to several interesting examples, including arithmetic circuits and power series.
In particular, the ring R of a function is the set of functions from X to R such that x commutes with y: xy=xx for all y in X. There are many other similar rings that are not necessarily associated with functions, for example the ring of integers modulo composites.
There is a great deal of research in this field, which is closely related to the study of commutative rings. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry.
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