  We analyze the stable isomorphism type of polynomial rings on degree 1
generators as modules over the sub-algebra A(1) = <Sq^1, Sq^2> of the mod 2
Steenrod algebra. Since their augmentation ideals are Q_1-local, we do this by
studying the Q_i-local subcategories and the associated Margolis localizations.
The periodicity exhibited by such modules reduces the calculation to one that
is finite. We show that these are the only localizations which preserve tensor
products, by first computing the Picard groups of these subcategories and using
them to determine all idempotents in the stable category of bounded-below
A(1)-modules. We show that the Picard groups of the whole category are detected
in the local Picard groups, and show that every bounded-below A(1) -module is
uniquely expressible as an extension of a Q_0-local module by a Q_1-local
module, up to stable equivalence.
  Applications include correct, complete proofs of Ossa's theorem, applications
to Powell's work describing connective K-theory of classifying spaces of
elementary abelian groups in functorial terms, and Ault's work on the hit
problem.
