  In this paper we show that each non-zero ideal of a twisted generalized Weyl
algebra (TGWA) $A$ intersects the centralizer of the distinguished subalgebra
$R$ in $A$ non-trivially. We also provide a necessary and sufficient condition
for the centralizer of $R$ in $A$ to be commutative, and give examples of TGWAs
associated to symmetric Cartan matrices satisfying this condition. By imposing
a certain finiteness condition on $R$ (weaker than Noetherianity) we are able
to make an Ore localization which turns out to be useful when investigating
simplicity of the TGWA. Under this mild assumption we obtain necessary and
sufficient conditions for the simplicity of TGWAs. We describe how this is
related to maximal commutativity of $R$ in $A$ and the (non-) existence of
non-trivial $\Z^n$-invariant ideals of $R$. Our result is a generalization of
the rank one case, obtained by D. A. Jordan in 1993. We illustrate our theorems
by considering some special classes of TGWAs and providing concrete examples.
