  We prove that within a natural class of E_3-algebras, the graded Tor group
induced by a span of E_3-algebra maps carries a functorial commutative graded
algebra structure generalizing the classical structure when the algebras are
genuine commutative differential graded algebras.
  As a topological corollary, Munkholm's Eilenberg--Moore collapse result for
pullbacks of spaces with polynomial cohomology is enhanced to a ring
isomorphism. New applications compute the Borel cohomology rings of homogeneous
spaces and singular cohomology rings of biquotients with minimal restrictions
on coefficient rings; existing results on the singular cohomology rings of free
loop spaces and (generalized) homogeneous spaces are respectively recovered and
strengthened.
