We describe in terms of generators and relations the ring structure of the R O ( C 2 ) RO(C_2) -graded C 2 C_2 -equivariant stable stems π ⋆ C 2 \pi _\star ^{C_2} modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational C 2 C_2 -equivariant sphere π ⋆ C 2 ( S Q ) \pi _\star ^{C_2}(\mathbb {S}_\mathbb {Q}) .