  We give a proper definition of the multiplicative structure of the following
rings: the Cox ring of invertible sheaves on a general algebraic stack; and the
Cox ring of rank one reflexive sheaves on a normal and excellent algebraic
stack. We show that such Cox rings always exist and establish their
(non-)uniqueness in terms of an Ext-group. Moreover, we compare our definition
with the classical construction of a Cox ring on a variety. Finally, we give an
application to the theory of Mori dream stacks.
