Geodesic
Multivectors of rank 2 over fields and commutative rings
Sbornik. Mathematics, Tome 193 (2002) no. 5, pp. 709-725

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In this paper the following question is studied: when can a homogeneous element of a Grassmann algebra be represented as the sum of two decomposable elements? For an exterior algebra over a field necessary and sufficient conditions of such a representation are obtained, over an arbitrary integral domain several necessary conditions, and over Krull rings also several sufficient conditions. In particular, it is established that the only rings such that the verification of 2-decomposability is carried out in the same way as over fields are the fields, that is, there are no “2-Plucker” rings. 
@article{SM_2002_193_5_a4,
     author = {G. B. Kleiner},
     title = {Multivectors of  rank 2 over fields and  commutative rings},
     journal = {Sbornik. Mathematics},
     pages = {709--725},
     publisher = {mathdoc},
     volume = {193},
     number = {5},
     year = {2002},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2002_193_5_a4/}
}
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G. B. Kleiner. Multivectors of  rank 2 over fields and  commutative rings. Sbornik. Mathematics, Tome 193 (2002) no. 5, pp. 709-725. http://geodesic.mathdoc.fr/item/SM_2002_193_5_a4/