A vector space 
with a ring structure and a vector
norm such that for all 
,
 |
If 
has a multiplicative
identity 1, it is also required that 
The field of real numbers 
is a normed ring with respect to the
absolute value, and the field
of complex numbers 
is a normed ring with respect to the modulus.
In both cases, the above inequality is actually an
equality. More general examples are the ring
of real square matrices
with the matrix norm and the ring
of real polynomials with a polynomial
norm.
