  We identify largest ideals in Leavitt path algebras: the largest locally
left/right artinian (which is the largest semisimple one), the largest locally
left/right noetherian without minimal idempotents, the largest exchange, and
the largest purely infinite. This last ideal is described as a direct sum of
purely infinite simple pieces plus purely infinite non-simple and
non-decomposable pieces. The invariance under ring isomorphisms of these ideals
is also studied.
