Pseudo amount

A pseudo amount is a weakened variant of an amount .

definition

Be a unitary ring . A mapping into the nonnegative real numbers is called a pseudo amount if the following properties apply to all : ${\displaystyle R}$ ${\displaystyle |\cdot |\colon R\to \mathbb {R} _{\geq 0}}$ ${\displaystyle a,b\in R}$

(1) (definiteness) ${\displaystyle |a|=0\;\Leftrightarrow \;a=0}$

(2) ${\displaystyle |a-b|\leq |a|+|b|}$

(3) ( sub-multiplicativity ) ${\displaystyle |a\cdot b|\leq |a|\cdot |b|}$

Will (3) tighten too

(3a) (multiplicativity), ${\displaystyle |a\cdot b|=|a|\cdot |b|}$

so is an amount . ${\displaystyle |\cdot |}$

The pseudo amount is called non-Archimedean if ${\displaystyle |\cdot |}$

(4) ${\displaystyle |a+b|\leq \max(|a|,|b|)}$

applies.

properties

• Always apply to a pseudo amount

${\displaystyle |{-a}|=|a|}$

and

${\displaystyle |a+b|\leq |a|+|b|}$ (Triangle inequality).

• The following always applies to a pseudo amount and even applies to an amount . ${\displaystyle |1|\geq 1}$ ${\displaystyle |1|=1}$
• 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

${\displaystyle {\begin{array}{llll}d\;\colon &R\times R&\to \mathbb {R} _{\geq 0}\\&d(a,b)&\mapsto |a-b|\end{array}}}$

defines the metric induced by the pseudo amount . It is an ultrametric if that is not Archimedean. ${\displaystyle |\cdot |}$

Examples

Let be a unitary ring with pseudo amount. ${\displaystyle (R,|\cdot |)}$

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. ${\displaystyle R[X]}$ ${\displaystyle R[X_{1},\ldots ,X_{n}]}$

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 . ${\displaystyle R^{n\times n}}$ ${\displaystyle 1\leq p\leq 2}$