Section 7.3
Domains and fields
Although some of the definitions and results presented in this and the following sections are valid for general rings, we concentrate on commutative rings. So, from now on, unless the contrary is stated explicitly, all rings will be considered to be commutative. In particular, let R be a commutative ring.
Definitions
- An element x of R is called a multiple of an
element y if there exists z
R such that x = y * z. - A zero divisor in
R is an element a of R for which there exists
b R\{0} with a * b = 0.
- A ring R without nonzero zero divisors is called a domain.
If x is a multiple of y, then y is also called
a
divisor of x.
Theorem
- A zero divisor is never invertible.
-
R is a domain if and only if, for all a, b R,
ab = 0 implies a = 0 or b = 0.
Clearly, a subring of a domain is a domain.