every ring is an integer algebra

Let $R$ be a ring. Then $R$ is also an algebra over the ring of integers if we define the action of $\mathbb{Z}$ on $R$ by the following rules:

$$0\cdot x=0$$

$$(n+1)\cdot x=n\cdot x+x$$

$$(-n)\cdot x=-(n\cdot x)$$

In other words, the action of a positive integer $n$ on $x$ is to add $x$ to itself $n$ times and the action of a negative integer $n$ on $x$ is to subtract $x$ to itself $n$ times.

Title	every ring is an integer algebra
Canonical name	EveryRingIsAnIntegerAlgebra
Date of creation	2013-03-22 14:47:47
Last modified on	2013-03-22 14:47:47
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	5
Author	rspuzio (6075)
Entry type	Example
Classification	msc 13B99
Classification	msc 20C99
Classification	msc 16S99
Related topic	GeneralAssociativity