The following always applies to a pseudo amount and even applies to an amount .
Every unitary ring with a magnitude is necessarily already an integrity ring (due to the multiplicativity, the zero divisor freedom of the real numbers is inherited on the ring).
The function
defines the metric induced by the pseudo amount . It is an ultrametric if that is not Archimedean.
Examples
Let be a unitary ring with pseudo amount.
Polynomial rings with pseudo amount
Then the polynomial algebras in one or more variables are themselves unitary rings (with the polynomial multiplication). The 1- pseudo norm is a pseudo amount on these polynomial rings.
Die rings with pseudo amount
Analogously, the matrix algebras are again unitary rings (here with the matrix multiplication ). Here is even the p-pseudo norm for every real p with a pseudo amount on the matrix ring .