ideal of an algebra

left idealright idealtwo-sided ideal

Let $A$ be an algebra over a ring $R$.

Definition - A left ideal of $A$ is a subalgebra $I\subseteq A$ such that $ax\in I$ whenever $a\in A$ and $x\in I$.

Equivalently, a left ideal of $A$ is a subset $I\subset A$ such that

1. $x-y\in I$, for all $x,y\in I$.

2. $rx\in I$, for all $r\in R$ and $x\in I$.

3. $ax\in I$, for all $a\in A$ and $x\in I$

Similarly one can define a right ideal by replacing condition 3 by: $xa\in I$ whenever $a\in A$ and $x\in I$.

A two-sided ideal of $A$ is a left ideal which is also a right ideal. Usually the word ”” by itself means two-sided ideal. Of course, all these notions coincide when $A$ is commutative.

0.0.1 Remark

Since an algebra is also a ring, one might think of borrowing the definition of ideal from ring . The problem is that condition 2 would not be in general satisfied (unless the algebra is unital).