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)
On the structure of $GL\sb 2$ over stable range one rings. (English) Zbl 0702.20037
For a large class of rings R (including all commutative rings) Vaserstein has proved that, when $n\ge 3$, the subgroups of $GL\sb n(R)$ normalized by $E\sb n(R)$, the subgroup generated by the elementary matrices, are completely determined by the R-ideals. From the extremely complicated normal subgroup structure of $SL\sb 2({\bbfZ})$, where ${\bbfZ}$ is the ring of rational integers, it is clear that for a general commutative ring R the subgroups of $GL\sb 2(R)$, normalized by $E\sb 2(R)$, cannot be completely determined in this way by special subsets of R like, for example, R-ideals. For anything like Vaserstein’s result to extend to the case $n=2$ it would appear that the ring has to contain “sufficiently many units”. Costa and Keller have proved that, if A is a commutative $SR\sb 2$-ring containing 1/2, then the normal subgroups of $SL\sb 2(A)$ are completely determined by the A-ideals. (Fields and semi-local rings, for example, are $SR\sb 2$-rings.) In the present paper the authors have extended this result to all $SR\sb 2$-rings B containing 1/2. They prove that the subgroups of $GL\sb 2(B)$, normalized by $E\sb 2(B)$, are completely determined by special subsets of B called quasi-ideals. (In any commutative ring containing 1/2 any quasi-ideal is an ideal.)
Reviewer: A.W.Mason

MSC:
20H25Other matrix groups over rings
20E07Subgroup theorems; subgroup growth
20G35Linear algebraic groups over adèles and other rings and schemes
WorldCat.org
Full Text: DOI
References:
[1] Abe, E.: Chavalley groups over local rings. Tôhoku math. J. 21, 474-494 (1969) · Zbl 0188.07201
[2] An, J. -B.; Tang, X. -P.: The structure of symplectic groups over semilocal rings. Acta math. Sinica (N.S.) 1, No. 1, 1-15 (1985)
[3] Bass, H.: K-theory and stable algebra. Publ. math. IHES 22, 5-60 (1964) · Zbl 0248.18025
[4] Costa, D. L.: Zero-dimensionality and the GE2 of polynomial rings. Pure appl. Algebra 50, No. 3, 223-229 (1988) · Zbl 0654.20058
[5] Costa, D. L.; Keller, G. E.: On the normal subgroups of $SL(2,A)$. J. pure appl. Algebra 53, 201-227 (1988) · Zbl 0654.20051
[6] Chang, C. N.: The structure of the symplectic group over semi-local domains. J. algebra 35, 457-476 (1975) · Zbl 0326.20041
[7] Dickson, L. E.: Theory of linear groups in arbitrary fields. Trans. amer. Math. soc. 2, 363-394 (1901) · Zbl 32.0131.03
[8] Fine, B.; Newman, M.: The normal subgroup structure of the Picard group. Trans. amer. Math. soc. 302, No. 2, 769-786 (1987) · Zbl 0624.20031
[9] Gerasimov, V. N.: The unit group of free products. Mat. sb. 1, 42-65 (1987) · Zbl 0634.16003
[10] Goodearl, K. R.: Von Neumann regular rings. (1979) · Zbl 0411.16007
[11] Goodearl, K. R.; Menal, P.: Stable range one for rings with many units. J. pure appl. Algebra 54, 261-287 (1988) · Zbl 0653.16013
[12] Klingenberg, W.: Lineare gruppen über lokalen ringen. Amer. J. Math. 83, 137-153 (1961) · Zbl 0098.02303
[13] Kolotilina, L. Yu.; Vavilov, N. A.: Normal structure of the full linear group over semilocal ring. J. soviet math. 19, No. 1, 998-999 (1982) · Zbl 0485.20045
[14] Lacroix, N. H. J.: Two-dimensional linear groups over local rings. Canad. J. Math. 21, 106-135 (1969) · Zbl 0169.34404
[15] Lacroix, N. H. J.; Levesque, C.: Sur LES sous-groupes normaux de SL2 sur un anneau local. Canad. math. Bull. 26, No. 2, 209-219 (1983) · Zbl 0515.20032
[16] Magurn, A. W.; Vaserstein, L. N.: Pre-stabilization for K1 of Banach algebras. Linear algebra 95, 69-96 (1987) · Zbl 0629.20021
[17] Mason, A. W.: Anomalous normal subgroups of SL2K[x]. Quart. J. Math. 36, No. 143, 345-358 (1985) · Zbl 0578.20037
[18] Mason, A. W.: On GL2 of a local ring in which 2 is not a unit. Canad. math. Bull. 30, No. 2, 165-176 (1987) · Zbl 0589.20032
[19] Mcdonald, B. R.: GL2 of rings with many units. Comm. algebra 8, No. 9, 869-888 (1980) · Zbl 0436.20031
[20] Mcdonald, B. R.: Geometric algebra over local rings. (1976) · Zbl 0346.20027
[21] Menal, P.; Moncasi, Jaume: K1 of von Neumann regular rings. J. pure appl. Algebra 33, No. 3, 295-312 (1984) · Zbl 0541.16021
[22] Menal, P.; Moncasi, Jaume: On regular rings with stable range 2. J. pure appl. Algebra 24, 25-40 (1982) · Zbl 0484.16006
[23] P. Menal and L.N. Vaserstein, On subgroups of GL2 over Banach algebras and von Neumann regular rings which are normalized by elementary matrices, J. Algebra, to appear. · Zbl 0724.20034
[24] Menal, P.; Vaserstein, L. N.: On normal subgroups of E2 over non-commutative local rings. Math. ann. 285, 221-231 (1989) · Zbl 0662.20039
[25] Newman, M.: A complete description of normal subgroups of genus one of the modular group. Amer. J. Math. 86, 17-24 (1964) · Zbl 0122.03703
[26] Newman, M.: Free subgroups and normal subgroups of the modular group. Illinois J. Math. 8, 262-265 (1964) · Zbl 0123.02803
[27] Reiner, I.: Normal subgroups of the unimodular group. Illinois J. Math. 2, 142-144 (1958) · Zbl 0078.01701
[28] Reiner, I.: Subgroups of the unimodular group. Proc. amer. Math. soc. 12, 173-174 (1961) · Zbl 0097.01603
[29] Serre, J. -P.: Le problème des groupes de congruences pour SL2. Ann. of math. 92, 489-527 (1970) · Zbl 0239.20063
[30] Suslin, A. A.: On a theorem of cohn. J. soviet math. 17, No. 2, 1801-1803 (1981) · Zbl 0462.12008
[31] Tazhetdinov, S.: Subnormal structure of two-dimensional linear groups over local rings. Algebra and logic 22, No. 6, 707-713 (1983) · Zbl 0542.20028
[32] Tazhetdinov, S.: Subnormal structure of two-dimensional linear groups over rings that are close to fields. Algebra and logic 24, No. 4, 414-425 (1985) · Zbl 0587.20028
[33] Vaserstein, L. N.: Math. notes. 5, 141-148 (1969)
[34] Vaserstein, L. N.: On normal subgroups of gln over a ring. Lecture notes in mathematics 854, 456-465 (1981)
[35] Vaserstein, L. N.: Normal subgroups of the general linear groups over von Neumann regular rings. Proc. amer. Math. soc. 96, No. 2, 209-214 (1986) · Zbl 0594.16007
[36] Vaserstein, L. N.: Normal subgroups of the general linear groups over Banach algebras. J. pure appl. Algebra 41, 99-112 (1986) · Zbl 0589.20030
[37] Vaserstein, L. N.: Subnormal structure of the general linear groups over Banach algebras. J. pure appl. Algebra 52, 187-195 (1988) · Zbl 0653.20050
[38] Vaserstein, L. N.: Bass’s first stable range condition. J. pure appl. Algebra 34, 319-330 (1984) · Zbl 0547.16017
[39] Vaserstein, L. N.: An answer to the question of M. Newman on matrix completion. Proc. amer. Math. soc. 97, 189-196 (1986) · Zbl 0601.15011
[40] L.N. Vaserstein, On normal subgroups of GL2 over rings with many units, Compositio Math., to appear. · Zbl 0697.20040
[41] Wang, L. Q.; Zhang, Y. Z.: GL2 over full rings. Chinese ann. Math. ser. B 8, No. 4, 434-439 (1987) · Zbl 0638.20029