ideal of a Lie algebra
In ring theory, the ideals are the subobjects you can quotient by; they are absorbent, swallowing products with anything in the ring. Lie algebras have the same notion, and for the same reason: an ideal is the kind of subalgebra you are allowed to collapse to a point, producing a new, smaller Lie algebra. It is what makes the isomorphism theorems run.
An ideal of a Lie algebra L is a subspace I such that [x, y] lies in I whenever x lies in L and y lies in I — written compactly [L, I] is contained in I. Because the bracket is antisymmetric, there is no distinction between left and right ideals; every ideal is automatically two-sided. Every ideal is a subalgebra, but not conversely.
The point of ideals is the quotient: if I is an ideal then the quotient vector space L/I inherits a well-defined bracket [x + I, y + I] = [x, y] + I, making it a Lie algebra, and the projection L -> L/I is a Lie algebra homomorphism with kernel I. Conversely the kernel of any Lie algebra homomorphism is an ideal. Important examples are the center, the derived algebra [L, L], and the terms of the derived and lower central series.
In gl(n), the scalar matrices c.I form a 1-dimensional ideal (the center), and sl(n) — the trace-zero matrices — is an ideal of codimension 1, since trace([X, Y]) = 0 always puts commutators inside sl(n). The quotient gl(n)/sl(n) is the 1-dimensional abelian Lie algebra.
sl(n) as a codimension-1 ideal of gl(n), since commutators are traceless.
Simplicity, solvability, and semisimplicity are all phrased in terms of ideals: a simple Lie algebra is nonabelian with no proper nonzero ideal, while a semisimple one has no nonzero solvable ideal.