  Given a family of based CW-pairs
$(\underline{X},\underline{A})=\{(X;A)\}^m_{i=1}$ together with an abstract
simplicial complex $K$ with $m$ vertices, there is an associated based
CW-complex $Z(K;(\underline{X},\underline{A}))$ known as a generalized
moment-angle complex.
  The decomposition theorem of \cite{bbcg}, \cite{bbcg2} splits the suspension
of $Z(K; (\underline{X}, \underline{A}))$ into a bouquet of spaces determined
by the full sub-complexes of $K$. Thatdecomposition theorem is used here to
describe the ring structure for the cohomology of Z(K; (\underline{X},
\underline{A})). Explicit computations are made for families of suspension
pairs and for the cases where $X_i$ is the cone on $A_i$. These results
complement and generalize those of Davis-Januszkiewicz, Franz, Hochster as well
as Panov, and Baskakov-Buchstaber-Panov. Under conditions stated below, these
theorems also apply for generalized cohomology theories.
