ideal of an algebra
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.
$xy\in I$, 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 twosided ideal of $A$ is a left ideal which is also a right ideal. Usually the word ”” by itself means twosided 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).
Title  ideal of an algebra 

Canonical name  IdealOfAnAlgebra 
Date of creation  20130322 18:09:00 
Last modified on  20130322 18:09:00 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  6 
Author  asteroid (17536) 
Entry type  Definition 
Classification  msc 16D25 
Synonym  left ideal of an algebra 
Synonym  right ideal of an algebra 
Synonym  twosided ideal of an algebra 