Ring Ideals

Definition

Given a ring $(R, +, \cdot )$ with additive group $(R, +)$. A subset $I \subset R$ is called a left ideal if

$(I, +)$ is a subgroup of $(R, +)$
$r\cdot x \in I$ for every $r \in R$ and $x \in I$

Similarly, it is called a left ideal if (2) is replaced with $x \cdot r \in I$ for every $r \in R$ and $x \in I$. If $I$ is an ideal if it is both a left and right ideal.