The ring
Elements with multiplicative inverses are terms units of a ring. In
The kernel of a homomorphism
Closure under multiplication is formalized by a subset

is a subgroup of 
and implies
Any ring with
Homomorphisms
Let
For
Adjunction of elements
Notation:
Let
Fields and integral domains
Integral domains are fields without zerodivisors, and fields have the additional property that every element has a multiplicative inverse.
No zerodivisors: satisfies the cancellation law that if
Fields are characterized by having exactly two ideals:
Maximal ideals
Nullstellansatz
Given three functions in two variables:
We can write
Algebraic Geometry
The set of solutions of
Two polynomials of degree
Unique Factorization Domain and Principal Ideal Domain
However, a UFD allows for associate factorizations: since the units of
Irreducible elements in a UFD are prime, but irreducible and prime are not equivalent in the general case: in
An integral domain in which all ideals are principal (generated by a single element) is termed a principal ideal domain.
Gauss Primes
The ring of Gauss Integers given by

The prime number
is either a Gauss prime, or the product of complex conjugate Gaussian primes. 
Let
be a Guass prime. Then, is either a prime integer or the square of a prime integer.
Algebraic numbers
A number
Algebraic integers in the quadratic field

If
, then 
If
, then