Two-sided ideal

Definition

Let $R$ be a ring. A two-sided ideal of $R$ is a subset $I \subseteq R$ with these properties.

1. $I$ is a subgroup of the additive group $(R, +)$.
2. For all $r \in R$ and $a \in I$, $r a \in I$ and $a r \in I$.

So a two-sided ideal is a left ideal that is also a right ideal.

Remarks

In a commutative ring, left ideals, right ideals and two-sided ideals are the same. The kernel of every ring homomorphism is a two-sided ideal. The quotient ring $R/I$ exists exactly when $I$ is a two-sided ideal. In the ring of $n \times n$ matrices over a field, the only two-sided ideals are $\{0\}$ and the whole ring.

Explanations

L1 human

In a noncommutative ring, $r a$ and $a r$ can be different. A two-sided ideal must absorb products from both sides. This is the condition that makes the quotient ring work.