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