Rings with identity

Abbreviation: Rng₁

Definition

A ring with identity is a structure R = (R, + ,−,0,·,1 ) of type (2,1,0,2,0) such that
(R, + ,−,0,·) is a ring and
1 is an identity for ·:   x·1 = x  and  1·x = x.

Morphisms

Let R and S be rings with identity. A morphism from R to S is a function h : R→S that is a homomorphism: h(x + y) = h(x) + h(y)  and  h(x·y) = h(x)·h(y)  and  h(1) = 1.

Remark: It follows that h(0) = 0 and h(−x) = −h(x).

Some results

0 is a zero for ·: 0·x = 0 and x·0 = 0.

Examples

(Z, + ,−,0,·,1), the ring of integers with addition, subtraction, zero, multiplication, and one.

Properties

Finite members

Subclasses

Superclasses