  This paper generalizes a result of Lynn on the "degree" of an equivariant
cohomology ring $H^*_G(X)$. The degree of a graded module is a certain
coefficient of its Poincar\'{e} series, and is closely related to multiplicity.
In the present paper, we study these commutative algebraic invariants for
equivariant cohomology rings. The main theorem is an additivity formula for
degree: $$deg(H^*_G(X)) = \sum_{[A,c] \in
\mathcal{Q'}_{max}(G,X)}\frac{1}{|W_G(A,c)|} \deg(H^*_{C_G(A,c)}(c)).$$ We also
show how this formula relates to the additivity formula from commutative
algebra, demonstrating both the algebraic and geometric character of the degree
invariant.
