Relation between rank and multiplicative complexity of a bilinear form over a commutative Noetherian ring

The concept of multiplicative complexity of a bilinear form is introduced for a commutative Noetherian ring. Rings are described for which the multiplicative complexity coincides with the rank for all forms. It is shown that for regular rings of dimension $\geqslant3$ the multiplicative complexity can exceed the rank by an arbitrarily large number.