  We use modular invariant theory to establish a complete set of relations of
the mod $p$ homology of $\{QS^k\}_{k\geq0}$, for $p$ odd, as a ring object in
the category of coalgebras (also known as a coalgebraic ring or a Hopf ring).
We also describe the action of the mod $p$ Dyer-Lashof algebra as well as the
mod $p$ Steenrod algebra on the coalgebraic ring.
