Let $(X,\sigma)$ be a symplectic space admitting a complex structure and let $R(X,\sigma)$ be the corresponding resolvent algebra, i.e. the $C^*$-algebra generated by the resolvents of selfadjoint operators satisfying canonical commutation relations associated with $(X,\sigma)$. In previous work this algebra was shown to provide a convenient framework for the analysis of quantum systems. In the present article its mathematical properties are elaborated with emphasis on its ideal structure. It is shown that $R(X,\sigma)$ is always nuclear and, if $X$ is finite dimensional, also of type I (postliminal). In the latter case dim$(X)$ labels the isomorphism classes of the corresponding resolvent algebras. For $X$ of arbitrary dimension, principal ideals are identified which are the building blocks for all other ideals. The maximal and minimal ideals of the resolvent algebra are also determined.