  Necessary and sufficient conditions for when every non-zero ideal in a
relative Cuntz-Pimsner ring contains a non-zero graded ideal, when a relative
Cuntz-Pimsner ring is simple, and when every ideal in a relative Cuntz-Pimsner
ring is graded, are given. A "Cuntz-Krieger uniqueness theorem" for relative
Cuntz-Pimsner rings is also given and condition (L) and condition (K) for
relative Cuntz-Pimsner rings are introduced.
