  Demailly, Ein and Lazarsfeld \cite{DEL} proved the subadditivity theorem for
multiplier ideals, which states the multiplier ideal of the product of ideals
is contained in the product of the individual multiplier ideals, on
non-singular varieties. We prove that, in two-dimensional case, the
subadditivity theorem holds on log-terminal singularities. However, in higher
dimensional case, we have several counter-examples. We consider the
subadditivity theorem for monomial ideals on toric rings, and construct a
counter-example on a three-dimensional toric ring.
