Commutative ring/Ideal/Definition

Ideal

A subset ${}{\mathfrak {a}}$ of a commutative ring ${}R$ is called an ideal, if the following conditions are fulfilled:

1. ${}0\in {\mathfrak {a}}$.
2. For all ${}a,b\in {\mathfrak {a}}$, we have ${}a+b\in {\mathfrak {a}}$.
3. For all ${}a\in {\mathfrak {a}}$ and ${}r\in R$, we have ${}ra\in {\mathfrak {a}}$.