Tautological rings

The tautological ring $R^{\ast }({\overline{\mathcal{M}}}_{g,n})$ is the subring of the the Chow ring $A^{\ast }({\overline{\mathcal{M}}}_{g,n})$ which is meant to contain all of the natural geometric information. Faber and Pandharipande gave the following elegant formulation (which is equivalent to previous definitions). There are forgetful morphisms ${\overline{\mathcal{M}}}_{g,n}\to {\overline{\mathcal{M}}}_{g,n-1}$ and gluing morphisms ${\overline{\mathcal{M}}}_{g,n+1}\times {\overline{\mathcal{M}}}_{h,m+1}\to {\overline{\mathcal{M}}}_{g+h,n+m}$ and ${\overline{\mathcal{M}}}_{g,n+2}\to {\overline{\mathcal{M}}}_{g+1,n}$. The system of tautological rings is then the smallest system of $\mathbb{Q}$-subalgebras of the Chow rings which is closed under the gluing and pushforward maps and which contains all of the Witten classes ${\psi }_i$. Tautological rings for the uncompactified moduli space and its partial compactifications are defined by restriction.