# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Operations on orbits of unimodular vectors. (English) Zbl 0592.20053
Let A be an associative commutative ring with 1. Then $GL\sb nA$ and $E\sb nA$ act on the unimodular rows $Um\sb nA\subset A\sp n$. The orbit set $Um\sb nA/E\sb nA$ is the set of unimodular rows modulo the addition operations over A. The author introduces an action of the multiplicative semigroup of the integers on the set $Um\sb nA/E\sb nA$ provided $n\ge 3$. He shows that taking the m-power of an entry in $(a\sb 1,...,a\sb n)\in Um\sb nA$ is a well-defined operation $\psi\sb m$ on $Um\sb nA/E\sb nA$, $n\ge 3$.
Reviewer: E.W.Ellers

##### MSC:
 20G35 Linear algebraic groups over adèles and other rings and schemes
##### Keywords:
elementary matrices; unimodular rows; action
Full Text:
##### References:
 [1] Van Der Kallen, W.: A group structure on certain orbit sets of unimodular rows. J. algebra 82, 363-397 (1983) · Zbl 0518.20035 [2] Spanier, E.: Borsuk’s cohomotopy groups. Ann. math. 50, No. 1, 203-245 (1949) · Zbl 0032.12402 [3] Suslin, A. A.: Orbits of SL3 and quadratic forms. (1977) [4] Suslin, A. A.: On stably free modules. Mat. sb. 102, No. 4, 537-550 (1977) · Zbl 0389.13002 [5] Suslin, A. A.: Reciprocity laws and the stable range in polynomial rings. Izv. akad. Nauk SSSR. Ser. mat. 43, No. 6, 1394-1425 (1979) · Zbl 0428.13007 [6] Suslin, A. A.: Mennicke symbols and their applications in K-theory of fields. Lecture notes in math., 334-356 (1982) [7] Suslin, A. A.: Homology of gln, characteristic classes, and Milnor K-theory. Lecture notes in math., 357-375 (1984) [8] Swan, R. G.; Towber, J.: A class of projective modules which are nearly free. J. algebra 36, 427-434 (1975) · Zbl 0322.13009 [9] Vaserstein, L. N.; Suslin, A. A.: Math. USSR izv.. 10, No. 5, 937-1001 (1975) [10] Vaserstein, L. N.: Stable rank of rings and dimensionality of topological spaces. Funct. anal. Appl. 5, 102-110 (1971) · Zbl 0239.16028 [11] Vaserstein, L. N.: On K1-theory of topological paces, preprint, 1983. (1986)