# 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 EveryRingIsAnIntegerAlgebra 2013-03-22 14:47:47 2013-03-22 14:47:47 rspuzio (6075) rspuzio (6075) 5 rspuzio (6075) Example msc 13B99 msc 20C99 msc 16S99 GeneralAssociativity