# Multivectors of rank 2 over fields and commutative rings

## Abstract

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.

- Authors:

- Central Economics and Mathematics Institute, Russian Academy of Sciences, Moscow (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21205698

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 193; Journal Issue: 5; Other Information: DOI: 10.1070/SM2002v193n05ABEH000652; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; COMMUTATION RELATIONS; INTEGRALS; VERIFICATION

### Citation Formats

```
Kleiner, G B.
```*Multivectors of rank 2 over fields and commutative rings*. United States: N. p., 2002.
Web. doi:10.1070/SM2002V193N05ABEH000652; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Kleiner, G B.
```*Multivectors of rank 2 over fields and commutative rings*. United States. doi:10.1070/SM2002V193N05ABEH000652; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Kleiner, G B. Sun .
"Multivectors of rank 2 over fields and commutative rings". United States. doi:10.1070/SM2002V193N05ABEH000652; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21205698,
```

title = {Multivectors of rank 2 over fields and commutative rings},

author = {Kleiner, G B},

abstractNote = {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.},

doi = {10.1070/SM2002V193N05ABEH000652; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 5,

volume = 193,

place = {United States},

year = {2002},

month = {6}

}

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