The reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems

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