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 "ideal" by itself means two-sided ideal. Of course, all these notions coincide when $A$ is commutative.

Remark

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