  Let $K$ be a field, and $A=K[a_1,\ldots ,a_n]$ a solvable polynomial algebra
in the sense of [K-RW, {\it J. Symbolic Comput.}, 9(1990), 1--26]. Based on the
Gr\"obner basis theory for $A$ and for free modules over $A$, an elimination
theory for left ideals of $A$ and an elimination theory for submodules of free
$A$-modules are established.
