Geodesic
Relation between rank and multiplicative complexity of a bilinear form over a commutative Noetherian ring
Zapiski Nauchnykh Seminarov POMI, Algebraic numbers and finite groups, Tome 86 (1979), pp. 66-81

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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. 
@article{ZNSL_1979_86_a7,
     author = {D. Yu. Grigor'ev},
     title = {Relation between rank and multiplicative complexity of a bilinear form over a commutative {Noetherian} ring},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {66--81},
     publisher = {mathdoc},
     volume = {86},
     year = {1979},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1979_86_a7/}
}
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D. Yu. Grigor'ev. Relation between rank and multiplicative complexity of a bilinear form over a commutative Noetherian ring. Zapiski Nauchnykh Seminarov POMI, Algebraic numbers and finite groups, Tome 86 (1979), pp. 66-81. http://geodesic.mathdoc.fr/item/ZNSL_1979_86_a7/