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.