In abstract algebra a rng is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. The term "rng" is meant to suggest that it is a "ring" without an "identity element", i. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In higher mathematics, algebraic structure is a loosely-defined phrase referring to the mathematical objects traditionally studied in the field of abstract algebra: sets with operations. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ...

(Some authors do not require that rings have a multiplicative identity, and for these authors the terms "rng" and "ring" are synonymous.)

Examples

Of course all rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng. In mathematics, any integer (whole number) is either even or odd. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

Rngs often appear naturally in functional analysis when linear operators on infinite-dimensional vector spaces are considered. Take for instance any infinite-dimensional vector space V and consider the set of all linear operators f : V → V with finite rank (i.e. dim f(V) < ∞). Together with addition and composition of operators, this is a rng, but not a ring. Another example is the rng of all real sequences that converge to 0, with component-wise operations. Finally, the real-valued continuous functions with compact support defined on some topological space, together with pointwise addition and multiplication, form a rng; this is not a ring unless the underlying space is compact. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, the dimension of a vector space V is the cardinality (i. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... Limit of a sequence is one of the oldest concepts in mathematical analysis. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...

Properties

Ideals and quotient rings can be defined for rngs in the same manner as for rings. The ideal theory of rngs is complicated by the fact that a rng, unlike a ring, need not contain any maximal ideals. Many theorems of ring theory are false for rngs. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, more specifically in ring theory a maximal ideal is a special kind of ideal which is in some sense maximal, that is not contained in any other non-trivial ideal of the ring. ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ...

Every rng R can be turned into a ring R^ by adjoining an identity element: set R^ = R × Z and define addition and multiplication in R^ by In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...

(r₁,n₁) + (r₂,n₂) = (r₁ + r₂, n₁+n₂)

and

(r₁,n₁) · (r₂,n₂) = (r₁r₂+n₂r₁+n₁r₂, n₁n₂)

The multiplicative identity of R^ is (0,1). R is a two-sided ideal in R^, so we can say:

Every rng is an ideal in some ring, and every ideal in some ring is a rng.

Rng-homomorphisms are defined just like ring homomorphisms, except that the requirement f(1)=1 is dropped. In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ...

Let j : R → R^ be the natural rng-homomorphism defined by j(r) = (r, 0). This map has the following universal property: given any ring S and any rng-homomorphism f : R → S, there exists a unique ring homomorphisms g : R^ → S such that f = gj. In a sense then, R^ is "the most general" ring containing R. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

The preceding paragraph can also be formulated in category theory. If we denote the category of all rings and ring homomorphisms by Ring and the category of all rngs and rng-homomorphisms by Rng, then we have an obvious forgetful functor F : Ring → Rng. The construction of R^ given above yields a left adjoint of F. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... A forgetful functor is a type of functor in mathematics. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ...