The complexity of equivalence for commutative rings

We study the deterministic time complexity of the equivalence problems for formulas and for straight-line programs on commutative rings. A general theorem is presented, that yields sufficient conditions on a commutative ring, for these problems for the ring to require “essentially as much deterministic time as the set of satisfiable 3CNF formulas”. As corollaries of this theorem, we characterize the deterministic time complexity of these two equivalence problems, for all finite commutative rings and for all commutative unitary rings of zero or prime characteristic