Rings Of Stable Rank Research Articles

Commutator width of Chevalley groups over rings of stable rank 1

Abstract It is shown that each element of the elementary Chevalley group of rank greater than 2 over a ring of stable rank 1 can be expressed as a product of few commutators.

UNITRIANGULAR FACTORIZATION OF TWISTED CHEVALLEY GROUPS

Unitriangular factorization is a presentation of a linear group as a product of unipotent radicals of a Borel subgroup and its opposite. Whether this decomposition is known for Chevalley groups over rings of stable rank 1 and some Dedekind rings of arithmetic type, the case of twisted groups has been studied only over finite fields. In the present paper we give a much simpler proof for twisted groups over finite fields and the field of complex numbers.

SL2-factorizations of Chevalley groups

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental SL2-factorizations of length 5N, where N is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov, it follows that the elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length 4N. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to 3N. On the other hand, for SL(n, R), over a Bezout ring R, we further improve the bound to 2N = n 2-n. Bibliography: 25 titles.

Unitriangular factorizations of chevalley groups

Lately, the following problem attracted a lot of attention in various contexts: find the shortest factorization G = UU - UU - …U ± of a Chevalley group G = G(Φ, R) in terms of the unipotent radical U = U(Φ, R) of the standard Borel subgroup B = B(Φ, R) and the unipotent radical U - = U -(Φ, R) of the opposite Borel subgroup B - = B - (Φ, R). So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits the unitriangular factorization G = UU - UU - U of length 5. Their proof invokes deep analytic and combinatorial tools. In the present paper, we notice that from the work of Bass and Tavgen one immediately gets a much more general results, asserting that over any ring of stable rank 1 one has the unitriangular factorization G = UU - UU - of length 4. Moreover, we give a detailed survey of traingular factorizations, prove some related results, discuss prospects of generalization to other classes of rings, and state several unsolved problems. Another main result of the present paper asserts that, in the assumption of the Generalized Riemann’s Hypothesis, Chevalley groups over the ring $$ \mathbb{Z}\left[ {\frac{1}{p}} \right] $$ admit the unitriangular factorization G = UU - UU - UU - of length 6. Otherwise, the best length estimate for Hasse domains with infinte multiplicative groups that follows from the work of Cooke and Weinberger, gives 9 factors. Bibliography: 67 titles.

On special congruence subgroups of symplectic groups

In this paper, it is proved that every special congruence subgroup SSp(V, I) of the symplectic group Sp(V (R)), where R is a ring of stable rank 1 with invertible element 2 and dim V (R) ≥ 4, is generated by the symplectic transvections belonging to this subgroup. This result is used to obtain the complete description of the normal subgroups of Sp(V (R)).

Elementary Reduction of Matrices over Right 2-Euclidean Rings

We introduce a concept of noncommutative (right) 2-Euclidean ring. We prove that a 2-Euclidean ring is a right Hermite ring, a right Bezout ring, and a GE n -ring. It is shown that an arbitrary right unimodular string of length not less than 3 over a right Bezout ring of stable rank possesses an elementary diagonal reduction. We prove that a right Bezout ring of stable rank 1 is a right 2-Euclidean ring.

Study of formal triangular matrix rings

In this paper we carry out a systematic study of various ring theoretic properties of formal triangular matrix rings. Some definitive results are obtained on these rings concerning properties such as being respectively left Kasch, right mininjective, clean, potent, right PF or a ring of stable rank ≤ n. The concepts of a strong left Kasch ring, a strong right mininjective ring are introduced and it is determined when the triangular matrix rings are respectively strong left Kasch or strong right mininjective. It is also proved that being strong left Kasch or strong right mininjective are Morita invariant properties.

On stable geometries

Within the concept of projective lattice geometry we are considering the class of stable geometries which have also been introduced in [14]. The investigation of their basic properties will result in fundamental structure theorems which especially give a lattice-geometric characterization of free left modules of rank ≥6 over proper right Bezout rings of stable rank 2. This yields a proper generalization of previous results of ours.

Rings of Stable Rank 2 are Barbilian Rings

Let MR a free unitary modul. A non-empty subset B of MR is by definition a Barbilian set iff B* ≔ F ⊂ B ¦ F is a free generating system of MRand #F > 1 satisfies: (B1) Each u1 ϵ B may be completed to 1, u2, … ϵ B* (B1) 1, u2, … ϵ B* implies 1 + u2α, u2, … ϵ B* for all α ϵ R. For each Barbilian set B ⊂ MR holds: B ⊂ Bmax= ϵ MR ¦ u may be completed to a free generating system F of MRand #F > 1 and Bmax is a Barbilian set iff Bmax ≠ .

Commutators and companion matrices over rings of stable rank 1

We consider the group GL n A of all invertible n by n matrices over a ring A satisfying the first Bass stable range condition. We prove that every matrix is similar to the product of a lower and upper triangular matrix, and that it is also the product of two matrices each similar to a companion matrix. We use this to show that, when n ⩾3 and A is commutative, every matrix in SL n A is the product of two commutators.

Factorization of invertible matrices over rings of stable rank one

AbstractEvery invertible n-by-n matrix over a ring R satisfying the first Bass stable range condition is the product of n simple automorphisms, and there are invertible matrices which cannot be written as the products of a smaller number of simple automorphisms. This generalizes results of Ellers on division rings and local rings.

Projective Barbilian Spaces I

An axiomatic description of projective spaces of arbitrary finite dimension over rings of stable rank 2 is given, and homomorphisms between such spaces are described.

Normal structure of symplectic groups over a ring of stable rank 1

Normal structure of symplectic groups over a ring of stable rank 1

Projective planes over rings of stable rank 2

This paper gives an axiomatic characterization of projective planes over rings of stable rank 2. These rings, which are known from algebraic K-theory, beautifully reflect simple geometric properties. The basic relations in the plane are incidence and the neighbor relation. The axioms consist of a number of axioms expressing elementary relations between points and lines such as, e. g., the existence of a unique line joining any two non-neighboring points, and a couple of axioms ensuring the existence of transvections and dilatations.