ON THE NORMAL STRUCTURE OF THE

GENERAL LINEAR GROUP OVER A RING

ALEXEI STEPANOV

Chair of Mathematics II St.Petersburg Electrotechnical University 194356, Prof. Popov st. 5, St.Petersburg, Russia

Introduction

The study of the normal structure of the general linear group over general rings was initiated by H.Bass [Bs1], [Bs2] [Bs3] in mid 60-th. In particular he established a complete description of the normal subgroups of the quite general linear group over an arbitrary ring. Also H.Bass introduced a new notion of dimension of rings, the so called stable rank, and discovered that the principal structure results remain valid for groups whose degree is large with respect to the stable rank. J.Milnor initiated the study of the Steinberg group in this context [M]. Very important contributions in this direction are due to A.Bak, R.K.Dennis, W.van der Kallen, A.A.Suslin and L.N.Vaserstein.

The next major breakthrough was triggered by the works of J.S.Wilson [W], I.Z.Golubchik [Gl] and A.A.Suslin [S] who discovered that for commutative rings the above results remain valid starting with degree 3 independently of the dimension of the ground ring. Further results in this direction were obtained by Z.I.Borewich, I.Z.Golubchik, V.I.Kopeiko, A.V.Mikhalev, A.A.Suslin, G.Taddei, L.N.Vaserstein, N.A.Vavilov and many others. W. van der Kallen [vdK] and M.S.Tulenbaev [T] proved the analogous results for the Steinberg group (starting with dimension 4). Also it should be mentioned that V.N.Gerasimov constructed (in any dimension) an example of a ring such that the elementary subgroup is not normal in the general linear group (and therefore, the normal structure of the general linear group over such a ring is nonstandard). A complete review of this subject is given in the book by A.J.Hahn and O.T.O’Meara [HO’M] and in survey by N.A.Vavilov [Vv].

The most intriguing problem in the theory of linear groups over rings nowadays is to determine the exact class of rings where the behaviour of the general linear group and the

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The author gratefully acknowledges support of St.Petersburg Mayor’s Office for the grant for young scientists and the SFB 343 an der Universität Bielefeld.

The reseach described in this publication was made possible in part by Grant N. JHP100 from the International Science Foundation and Russian Goverment.

Steinberg group is standard, i.e. is almost the same as in stable situation described by H.Bass and J.Milnor. Partial results in this direction are obtained in the current article.

The rest of the article is organized as follows. In the first section the main notation and the notions of the standardness of the normal structure of the general linear group and the stable rank of a ring are introduced and their elementary properties are given. The next sections are devoted to:

— the normality of the elementary subgroup in the general linear group;

— the standard commutator formula;

— the standard distribution of the normal subgroups in the general linear group;

— the centrality of the extension St(n, R) → E(n, R), respectively.

An important idea of the article is that the standardness of the normal structure of GL(n, R) can be “lifted” modulo an ideal I if its stable rank is less than n. In particular, we will show that the standardness of the normal structure of GL(n, R) does not depend on the Jacobson radical Rad R of a ring R.

1. Main notation

(1.1) Let G be any group. For elements x, y ∈ G we denote by [x, y] = x^{−1}y^{−1}xy their commutator and by x^{y} = y^{−1}xy the conjugates of x by y. The double commutator [[x, y], z] is denoted by [x, y, z]. We write H ⩽ G to denote that H is a subgroup of G and H ◁ G to denote that H is a normal subgroup of G. For a subset X ⊆ G we denote by ⟨X⟩ ⩽ G the subgroup of G generated by X and by ⟨X⟩^{H} the smallest subgroup which contains X and is normalized by H. For two subgroups F, H ⩽ G we denote by [F, H] the corresponding relative commutator subgroup, generated by all commutators [f, h], f ∈ F, h ∈ H. (1.2) Let D and G′ be subgroups of G. Recall [St2] that a subgroup F of G′ is called D– perfect if [F, D] = F. For a given D–perfect subgroup F the largest subgroup C with the property [C, D] = F is called a generic subgroup corresponding to F (see [CK]). Clearly all subgroups from each sandwich F ⩽ … ⩽ C are normalized by D. Let us say that D is strongly polynormal in G′ if for any H ⩽ G′ the subgroup [H, D] is D–perfect. In other words, for any subgroup H ⩽ G′ there exist a unique D–perfect subgroup F such that H fits into the sandwich F ⩽ … ⩽ C where C is the corresponding generic subgroup.

In the sequel we will often use the following quite easy fact deriving from theorem 1 of [St2].

1.3. Proposition. Let D be a normal perfect subgroup of G, i.e. [D, D] = D ◁ G. Then D is strongly polynormal in G (and hence, in any subgroup of G).

(1.4) Let R be an associative ring with 1, GL(n, R) the general linear group of degree n over R. We always assume n ⩾ 3. As usual, for a matrix a ∈ GL(n, R) we denote by a_{ij} its matrix entry in position (i, j) and by a′_{ij} the corresponding element of its inverse a^{−1}. Quite often we will have to use the following notation for rows and columns of matrices a and a^{−1}: the i-th rows of this matrices will be denoted by a_{i∗} and a′_{i∗} while the i-th columns will be denoted by a_{∗i} and a′_{∗i} respectively. The identity matrix of degree n is denoted by e = e_{n} and the standard matrix units (i.e. matrices with 1 in position (i, j) and zeros elsewhere) by e^{ij}. The element of e in position (i, j) is denoted by δ_{ij} (Kroneker symbol).

An elementary transvection t_{ij}(ξ) is a matrix of the form t_{ij}(ξ) = e + ξe^{ij}, ξ ∈ R, 1 ⩽ i ≠ j ⩽ n. The subgroup E(n, R) of the general linear group GL(n, R), generated by all the elementary transvections t_{ij}(ξ) is called the elementary subgroup. Let α = (α_{1}, …, α_{n})^{T} be a column and β = (β_{1}, …, β_{j}) be a row of length n over a ring R. For shortness of notation in proofs we will use the following abbreviations:

t_{∗i}(β) = ∏_{j≠i} t_{ji}(β_{j}), t_{i∗}(α) = ∏_{j≠i} t_{ij}(α_{j}).

(1.5) Now recall the definitions of certain subgroups of GL(n, R) normalized by E(n, R). Each ideal I of a ring R defines the reduction homomorphism

ρ_{I} : GL(n, R) → GL(n, R/I),

such that the image of a matrix x = (x_{ij}) equals \overline{x} = (\overline{x}_{ij}). The kernel of the homomorphism ρ_{I} is denoted by GL(n, R, I) and is called the principal congruence subgroup of level I. The inverse image of C(n, R/I) under the reduction homomorphism ρ_{I} is denoted by C(n, R, I) and is called the full congruence subgroup of level I. The relative elementary subgroup of level I is a group

E(n, R, I) = ⟨t_{ij}(ξ) | ξ ∈ I, 1 ⩽ i ≠ j ⩽ n⟩^{E(n,R)}.

Note that a relative elementary subgroup is (if n ⩾ 3) always E(n, R)–perfect. In particular the group E(n, R) is perfect. (1.6) It turns out that in the standard situation the following assertions become true:

(1) E(n, R) is a normal subgroup of GL(n, R);

(2) for each ideal I ⩽ R the standard commutator formula

[GL(n, R, I), E(n, R)] = E(n, R, I)

holds

(3) for any subgroup H in GL(n, R) normalized by E(n, R) there exists a unique ideal

I ⩽ R such that

E(n, R, I) ⩽ H ⩽ C(n, R, I).

(4) The Steinberg group St(n, R) is a central extension of the elementary subgroup

E(n, R). We say that the normal structure of the general linear group is standard if items (1), (2) and (3) hold. (1.7) Suppose that E(n, R) ◁ GL(n, R). Then proposition 1.3 implies that standard commutator formula (1.6)(2) is equivalent to the formula

[C(n, R, I), E(n, R)] = E(n, R, I),

and that the elementary subgroup E(n, R) is strongly polynormal in GL(n, R). In this case the standardness of the normal structure of the general linear group means that only relative elementary subgroups are E(n, R)–perfect in GL(n, R). (1.8) The last observation proves that the following definition is natural. Let us say that the normal structure of a subgroup G′ ⩽ GL(n, R) is standard if the groups E(n, R, I) and only they are E(n, R)–perfect in G′. In (5.1) the standardness of the normal structure of the Steinberg group is defined in the same way. The following propositions collect some easy facts about the standard normal structure.

1.9. Proposition. Let D be a subgroup in G and φ : G ⟶ \overline{G} be a group homomorphism.

(i) The image of any D–perfect subgroup H is φ(D)–perfect. Conversely, for any φ(D)–perfect subgroup \overline{H} there exists a D–perfect subgroup H such that φ(H) = \overline{H}.

(ii) If D is strongly polynormal in G then φ(D) is strongly polynormal in \overline{G}.

1.10. Proposition.

(i) If E(n, R) is normal in GL(n, R) then standard normal structure of E(n, R) is equivalent to standard normal structure of GL(n, R).

(ii) Standard normal structure of E(n, R) (St(n, R)) implies standard normal structure of E(n, R/I) (respectively, St(n, R/I)) for any ideal I in R.

(1.11) Let R^{n} be a free right R-module consisting of all columns of height n with components in R. Following P.M.Cohn we denote free left R-module consisting of all rows of length n with components in R by ^{n}R. The standard bases of R^{n} and ^{n}R are denoted by f^{1}, …, f^{n} and e^{1}, …, e^{n} respectively. In other words, f^{i} is the i-th column of the identity matrix e of order n (specified by context), while e^{i} is the i-th row of e. (1.12) The notion of stable rank introduced by H.Bass [Bs2], [Bs3] will be used quite often through the article. We will use the notation and some results of L.N.Vaserstein [Vs1]. Recall that a row α = (α_{1}, …, α_{n}) ∈ ^{n}R is called unimodular if its elements generate the whole ring R as a right ideal. Let I be a (two-sided) ideal in R. A row α is called I– unimodular if it is unimodular and α_{i} − δ_{1i} ∈ I for all i = 1, …, n. A unimodular row α is called stable if there are λ_{1}, …, λ_{n−1} ∈ R such that the row α_{1} +α_{n}λ_{1}, …, α_{n−1} +α_{n}λ_{n−1} is also unimodular. (1.13) Let us say that the stable rank of a pair (R, I) equals d and write sr(R, I) = d if each I–unimodular row of length d + 1 is stable and there exist a nonstable I–unimodular row of length d. If there is no such d then sr(R, I) = ∞. The stable rank of a ring R (sr R) is the stable rank of the pair (R, R). It is known [Vs1] that any I–unimodular row of length greater than sr(R, I) is stable, and that sr(R, I) depends only on the ideal I (considered as a ring without the identity element). Recall [Vs1] that sr(R, J) = 1 if J ⩽ Rad R.

2. Normality of the elementary subgroup

Known methods of proof of the normality of the elementary subgroup in the general linear group are completely presented in [Vv], [StVv]. The following proposition is an acceleration of the idea of “decomposition of transvections” introduced in author’s thesis. It shows also that the condition of normality of E(n, R) in GL(n, R) is a condition of linear dependence of elements of a ring R.

2.1. Proposition. Let D = E(n, R) and G = GL(n, R). The following conditions are equivalent.

(i) D ◁ G.

(ii) For any a ∈ GL(n, R) the subgroup, generated by the matrices e + a′_{∗i}λa_{j∗}, i ≠ j,

λ ∈ R, contains D (symbols a_{j∗} and a_{∗j} were introduced in (1.4)). (iii) For any a ∈ GL(n, R), 1 ⩽ i ≠ j ⩽ n and λ ∈ R there exist a set of rows β^{k} ∈ ^{n}R

and a set of columns α^{k} ∈ R^{n} (k = 1, …, m) such that: β^{k}α^{k} = 0,

t_{ij}(λ) = ∏_{k=1}^{m}(e + α^{k}β^{k})

and each row β^{k}a contains at least one zero element.

Proof. If D ◁ G then D = D^{a}. Since D^{a} is generated by the matrices t_{ij}(λ)^{a} = e + a′_{∗i}λa_{j∗}, (i)⇒(ii). If one replaces a by a^{−1} in condition (ii), one will be able to represent any elementary transvection t_{ij}(λ) as a product of matrices e + a_{∗i_{k}}λ_{k}a′_{j_{k}∗}. It is easy to verify that the rows a′_{j_{k}∗} and the columns a_{∗i_{k}}λ_{k} satisfy the condition of item (iii).

Suppose now that the conditions of item (iii) hold. Then

t_{ij}(λ)^{a} = ∏_{k=1}^{m}(e + a^{−1}α^{k}β^{k}a).

Fix k and set γ = a^{−1}α^{k}, ω = β^{k}a. By assumption ω_{p} = 0 for some index p. Hence, by Whitehead–Vaserstein lemma

e + γω = t_{p∗}(γ_{p}ω) [t_{∗p}(γ), t_{p∗}(ω)] ∈ D.

Thus t_{ij}(λ)^{a} ∈ D, and D is normal in G.

The following theorem is devoted to lifting modulo an ideal of small stable rank. It seems that the converse of theorem (2.2) is not true. It is concerned with the fact that the general linear group grows in an unknown way under factorization of the ground ring. However, if an ideal I is contained in the Jacobson radical of R then the canonical map GL(n, R) → GL(n, R/I) is surjective, and hence, by (1.9)(ii), the converse of theorem 2.2 holds (cf. corollary 2.3).

2.2. Theorem. Let n ⩾ sr(R, I) + 1. If E(n, R/I) is normal in GL(n, R/I) then E(n, R) is normal in GL(n, R).

Proof. Obviously the subgroup

H = ρ_{I}^{−1}(E(n, R/I)) = E(n, R) · GL(n, R, I)

is normal in GL(n, R). Since n ⩾ sr(R, I) + 1, by the theorem of H.Bass [Bs2] the group K_{1}(n, R, I) is abelian. Hence, the subgroup E(n, R) = [H, H] is also normal.

2.3. Corollary. Suppose that an ideal I is contained in the Jacobson radical Rad R of a ring R. Then E(n, R) is normal in GL(n, R) if and only if E(n, R/I) is normal in GL(n, R/I).

3. The standard commutator formula

(3.1) Let I be a two-sided ideal of a ring R. Following A.Bak we denote by R ⋉ I their smash product, i.e. a subring of R × R consists of all elements (λ, μ) such that λ − μ ∈ I. Analogously, for a normal subgroup H of a group G we denote by G ⋉ H their semidirect product, i.e. the subgroup of G × G consisting from all pairs (a, b) such that ab^{−1} ∈ H.

The idea of the following proposition was often used by many authors (e.g. [Bk], [Vs2]). It shows that the standard commutator formula is almost equivalent to the normality of the elementary subgroup in the general linear group but over another ring.

3.2. Proposition. The subgroup E(n, R ⋉ I) is normal in GL(n, R ⋉ I) if and only if E(n, R) is normal in GL(n, R) and the standard commutator formula

[GL(n, R, I), E(n, R)] = E(n, R, I)

holds.

Proof. Clearly GL(n, R ⋉ I) is isomorphic to the semidirect product GL(n, R) ⋉ GL(n, R, I), and E(n, R ⋉ I) ≅ E(n, R) ⋉ E(n, R, I) under the same isomorphism. Let a ∈ E(n, R), c ∈ GL(n, R) and d ∈ GL(n, R, I). If E(n, R ⋉ I) is normal in GL(n, R ⋉ I) then

[(a, a), (c, cd)] = ([a, c], [a, d]^{c}[a, c]) ∈ E(n, R ⋉ I).

Hence, [a, c] ∈ E(n, R). It proves normality of E(n, R) in GL(n, R) and the inclusion [a^{c}, d^{c}] ∈ E(n, R, I). Since a^{c} and d^{c} are an arbitrary elements from E(n, R) and GL(n, R, I) respectively, we get the standard commutator formula. Conversely, if E(n, R) is normal in GL(n, R) and

[GL(n, R, I), E(n, R)] = E(n, R, I)

then

[(a, a), (c, cd)] ∈ E(n, R ⋉ I).

So the latter contains [D, GL(n, R ⋉ I)], where D = {(a, a) | a ∈ E(n, R)}. The Steinberg relations between elementary transvections show that in fact

E(n, R ⋉ I) = [D, GL(n, R ⋉ I)].

But the latter obviously is normal in GL(n, R ⋉ I).

The rest of this section is devoted to the proof of the analogue of theorem 2.2. The next lemma belongs to W.van der Kallen (see [CR], lemma 44.25).

3.3. Lemma. For any two-sided ideals I, J of a ring R

sr(R ⋉ I, J ⋉ I) ⩽ sr(R, J)

3.4. Corollary. Let I, J be two-sided ideals in R, n ⩾ sr(R, J) + 1 and \overline{I} = π_{J}(I) be the image of I in the ring \overline{R} = R/J. If E(n, R/J) is normal in GL(n, R/J) then the standard commutator formula

[GL(n, \overline{R}, \overline{I}), E(n, \overline{R})] = E(n, \overline{R}, \overline{I})

over the ring \overline{R} is equivalent to the analogous formula

[GL(n, R, I), E(n, R)] = E(n, R, I)

over R.

Proof. The direct implication follows immediately from (2.2), (3.2) and (3.3). The converse assertion in fact does not depend on the stable rank of J. Indeed, by the conditions the group

F = [GL(n, \overline{R}, \overline{I}), E(n, \overline{R})]

is contained in E(n, \overline{R}). But by proposition 1.3 it is E(n, \overline{R})–perfect. Therefore,

F = [GL(n, \overline{R}, \overline{I}) ∩ E(n, \overline{R}), E(n, \overline{R})] = ρ_{J}([GL(n, R, I + J), E(n, R) · GL(n, R, J)]).

But the right hand side of the last formula obviously coincides with

ρ_{J}(E(n, R, I)) = E(n, \overline{R}, \overline{I}).

4. Distribution of normal subgroups

The first part of this section is devoted to proof of theorem 4.4 that shows that under the standard commutator formula the standard distribution of normal subgroups (1.6)(3) follows from certain condition of linear dependence of elements of the ground ring (distinct from condition (iii) of proposition 2.1).We start with some lemmas. The proof of lemma 4.1 uses the standard technique. Its analogues can be found in [Bs3], [BV], [Gl], [StVv], [Vs2].

4.1. Lemma. Let F be a subgroup of GL(n, R) normalized by E(n, R) and c ∈ F. Suppose that F does not contain elementary transvections. If for some unimodular column α ∈ R^{n} and some indexes i, j ∈ {1, …, n} the elements α_{i} and (cα)_{j} equal zero then c is a central matrix. In particular c is central if it contains an invertible element or an element from the Jacobson radical of R.

Proof. Recall from [Bs3], [BV], [StVv] that if c_{ij} = 0 then either c is central or F contains a nontrivial elementary transvection. Let now c be a noncentral matrix and α satisfy the conditions of the lemma. Set β = cα and consider the matrix

d = ct_{∗i}(α)c^{−1}t_{∗i}(−α) = (e + βc′_{i∗})t_{∗i}(−α)

that belongs to F. Since β_{j} = 0, d_{jk} = δ_{jk} for all k ≠ i, and hence, d = e. It follows that βc′_{im} = 0 for any m ≠ i. Thus c′_{im} = 0 (because β is unimodular) which is impossible.

The assertion is obvious if c contains an invertible element. Suppose finally that c_{ij} ∈ Rad R. Consider the matrix a = [t_{ki}(1), c] ∈ F, where k ≠ i, j. Calculations shows that a_{mj} = δ_{mj} + c′_{mk}c_{ij}. In particular, the element a_{jj} is invertible, and hence, the matrix a is central. It follows that c′_{mk}c_{ij} = 0 for each m, and so c_{ij} = 0. Thus the matrix c is also central.

(4.2) For a matrix c ∈ GL(n, R) we denote by I(c) the smallest ideal in R such that c ∈ C(n, R, I), i.e. the smallest ideal containing the elements c_{ij}, c_{ii} − c_{jj} for all i ≠ j. By I^{⊥}(c) we denote an intersection of right and left annihilators of I(c).

The next lemma is a key step to the proof of theorem 4.4. It shows how to extract transvections from a matrix with a zero divizor in some place.

4.3. Lemma. Let F be a subgroup of GL(n, R) normalized by E(n, R), ξ ∈ R and c ∈ F. Suppose that F does not contain nontrivial elementary transvections and ξc_{ij} = 0 for some indexes i, j. Then ξ ∈ I^{⊥}(c).

Proof. Choose arbitrary λ ∈ R and an index k ≠ i. Consider the matrix

d = [t_{ki}(λξ), c] = t_{ki}(−λξ)(e + c′_{∗k}λξc_{i∗})

that belongs to F. Since ξc_{ij} = 0 then d_{mj} = δ_{mj} for all m ≠ k. Hence, by the last lemma the matrix d is central. One can note that in fact d = e because d_{jj} = 1 (the latter is true even if k = j). Thus c′_{lk}Rξ = 0 for all k ≠ i, l ≠ k. Then by the same arguments one can conclude that c commutes with transvections t_{lk}(λξμ) for all l ≠ k, λ, μ ∈ R. In particular, c_{kl}λξ = ξλc_{kl} = 0 for all l ≠ k, λ ∈ R.

Let now a = t_{pq}(−1)ct_{pq}(1) ∈ F, where p ≠ q. Since ξa_{ph} = 0 for all h distinct from p and q then as in the first paragraph of the proof we get ξλa_{pq} = 0. But

ξλa_{pq} = ξλ(c_{pq} + c_{pp} − c_{qq}) = ξλ(c_{pp} − c_{qq}).

Analogously, (c_{pp} − c_{qq})ξλ = 0, which completes the proof.

Let H be a subgroup of GL(n, R) normalized by E(n, R). One can note that the set of all elements ξ ∈ R such that t_{ij}(ξ) ∈ H for some (or, which is equivalent, for all) i ≠ j is an ideal in R. Recall from [BV] that it is called the ideal associated with H.

4.4. Theorem. Suppose that for any proper two-sided ideal I in R the following conditions hold:

(a) [GL(n, R, I), E(n, R)] = E(n, R, I);

(b) for any a ∈ GL(n, R) there exist b ∈ E(n, R/I) and α ∈ (R/I)^{n}, (α ≠ 0) such that α_{i} = (u^{b}α)_{j} = 0 for some i, j ∈ {1, …, n}, where u = ρ_{I}(a) is the image of a in GL(n, R/I).

Then the normal structure of GL(n, R) is standard.

Proof. Let I be an ideal associated with H and F = ρ_{I}(H) be the image of H in GL(n, R/I). If F contains a nontrivial elementary transvection then by the usual arguments (see [Bs3], [BV], [StVv]) we can lift it to H. But this contradicts the fact that all transvections from H vanish under the homomorphism ρ_{I}.

Suppose now that F does not contain nontrivial elementary transvections and a ∈ F. If a is noncentral then I^{⊥}(a) ≠ R/I. Let

\overline{R} = (R/I)/I^{⊥}(a), φ = ρ_{I^{⊥}(a)} : GL(n, R/I) → GL(n, \overline{R}) and \overline{a} = φ(a).

Since \overline{R} is a quotient ring of R and \overline{a} is the image of some matrix from H we can choose \overline{b} and \overline{α} satisfying the conditions of the theorem. Let α be a proimage of \overline{α} in (R/I)^{n}, b ∈ φ^{−1}(\overline{b}) and λ be an arbitrary element of the ring R/I. Set c = a^{b}, β = cα and consider the matrix

d = ct_{∗i}(αλ)c^{−1}t_{∗i}(−αλ) = (e + βλc′_{i∗})t_{∗i}(−αλ)

that belongs to F. Note that I(d) ⩽ I(c) = I(a). By condition (b) of the theorem the matrix φ(d) contains a zero element in position (j, k) for some k ≠ j, i.e. d_{jk} ∈ I^{⊥}(a) ⩽ I^{⊥}(d). Therefore, d_{jk}(R/I)d_{jk} = 0, and hence, d_{jk} ∈ Rad(R/I). Then by lemma 4.1 d belongs to the center of GL(n, R/I). Hence, ξ(R/I)c′_{im} = 0 for any m ≠ i, where ξ is an element of the row β which does not belong to I^{⊥}(c) (such an element exists because the image of β in \overline{R}^{n} is nonzero and I^{⊥}(c) = I^{⊥}(a) ). But this is impossible by lemma 4.3. The contradiction shows that a is central. Thus F is contained in the center of GL(n, R) and H ⩽ C(n, R, I).

In the rest of this section we will prove that the standardness of the normal structure of the general linear group can be “lifted” modulo an ideal of small stable rank. The proof of the next lemma can be extracted from [Bs2], and therefore will be omitted.

4.5. Lemma. If n ⩾ sr(R, J) + 1 then the normal structure of GL(n, R, J) is standard.

4.6. Theorem. Let n ⩾ sr(R, J)+1, and the normal structure of GL(n, R/J) be standard. Then the normal structure of GL(n, R) is also standard.

Proof. By corollary 3.4 the standard commutator formula holds in GL(n, R) for any ideal in R . In particular E(n, R) is normal in GL(n, R). Therefore, (see (1.7)) we have to prove that any E(n, R)–perfect subgroup H in GL(n, R) coincides with E(n, R, I). Let I be the ideal associated with H. Since the normal structure of GL(n, R/J) is standard,

ρ_{J}(H) = E(n, R/J, \overline{I})

for some ideal \overline{I} ⩽ R/I. As at the beginning of the proof of theorem 4.4 we can conclude that E(n, R/J, \overline{I}) does not contain a transvection t_{ij}(ξ) if ξ ∉ π_{J}(I). Thus \overline{I} = π_{J}(I). Hence, H ⩽ GL(n, R, J) · E(n, R, I). It is easy to see that there exists a subgroup F in GL(n, R, J) such that H = F · E(n, R, I). By lemma 4.5

E(n, R, I_{1}) ⩽ F ⩽ C(n, R, I_{1}) for some I_{1} ⩽ J.

Obviously I_{1} ⩽ I, and hence,

H ⩽ C(n, R, I_{1}) · E(n, R, I) ⩽ C(n, R, I).

Finally

H = [H, E(n, R)] ⩽ [C(n, R, I), E(n, R)] = E(n, R, I).

4.7. Corollary. If J ⩽ Rad R then standard normal structure of GL(n, R) is equivalent to standard normal structure of GL(n, R/J).

5. Centrality of K_{2}

In the last section we investigate relations between the centrality of the extension St(n, R) → E(n, R) and the standardness of the normal structure of the elementary subgroup E(n, R) and the Steinberg group St(n, R). Also we will show that the centrality of this extension is equivalent to the centrality of the extension St(n, R/J) → E(n, R/J) if n ⩾ sr(R, J) + 2. In the sequel we will give an example of a ring A for which the centrality of K_{2}(n, R) follows from theorem 5.4 and was not known so far. (5.1) Recall [Mi], [ST], [T] that K_{2}(n, R) is the kernel of the canonical projection π : St(n, R) → E(n, R) (the definition of the Steinberg group can be found in the cited works). We denote the generators of this group by x_{ij}(ξ). For an ideal I of a ring R we denote by St(n, R, I) the kernel of the natural epimorphism St(n, R) → St(n, R/I). In other words, St(n, R, I) is the smallest normal subgroup of St(n, R) containing all generators x_{ij}(ξ), i ≠ j, ξ ∈ I.

Obviously, any subgroup St(n, R, I) is St(n, R)–perfect. Let us say that the normal structure of the Steinberg group St(n, R) is standard if in the Steinberg group there are no other St(n, R)–perfect subgroups.

5.2. Proposition. The following conditions are equivalent.

(i) The normal structure of E(n, R) is standard and K_{2}(n, R) is central in St(n, R). (ii) The normal structure of St(n, R) is standard. (iii) Condition (i) or (ii) holds for all quotient rings R/I.

Proof. Let H be a St(n, R)–perfect subgroup. If the normal structure of the group E(n, R) is standard then π(H) coincides with one of the groups E(n, R, I). Hence,

H ⩽ St(n, R, I) · K_{2}(n, R) and H = [H, H] ⩽ St(n, R, I).

On the other hand, if ξ ∈ I then there is an element a ∈ K_{2}(n, R) such that ax_{12}(ξ) ∈ H. Then the formula

x_{32}(ξ) = [x_{31}(1), ax_{12}(ξ)] ∈ H

implies the inverse inclusion St(n, R, I) ⩽ H.

Conversely, since K_{2}(n, R) does not contain the generators x_{ij}(ξ), it is central by definition of the standard normal structure (5.1). The standardness of the normal structure of E(n, R) follows from (1.9). The equivalence (ii) ⟺ (iii) is obvious.

5.3. Lemma. If [GL(n, R, I), E(n, R)] = E(n, R, I) and K_{2}(n, R) is central in St(n, R) then K_{2}(n, R/I) is central in St(n, R/I).

Proof. Consider a commutative diagram

St(n, R) ⟶^{φ} St(n, R/I)
π↓  ↓
E(n, R) ⟶^{ρ_{I}} E(n, R/I)

Set H = φ^{−1}(K_{2}(n, R/I)) and H′ = [H, St(n, R)]. Clearly π(H) ⩽ GL(n, R, I). The commutator formula shows that π(H′) ⩽ E(n, R, I), i.e. H′ ⩽ St(n, R, I) · K_{2}(n, R). Since [H′, St(n, R)] = H′ (see (1.3)) the centrality of K_{2}(n, R) implies the inclusion H′ ⩽ St(n, R, I). Thus

[K_{2}(n, R/I), St(n, R/I)] = φ(H′) = {e}.

5.4. Theorem. Let n ⩾ sr(R, I) + 2. Then the centrality of K_{2}(n, R) is equivalent to the centrality of K_{2}(n, R/I).

Proof. By the corollary 3.4 we have the standard commutator formula

[GL(n, R, I), E(n, R)] = E(n, R, I).

Hence, by lemma 5.3 the centrality of K_{2}(n, R) implies the centrality of K_{2}(n, R/I). Conversely, if K_{2}(n, R/I) is contained in the center of the Steinberg group then

F = [K_{2}(n, R), St(n, R)] ⩽ Ker(φ) = St(n, R, I)

Since n ⩾ sr(R, I) + 2 we can repeat the proof of surjective stability for K_{2} from [ST] in the group St(n, R, I) and show that any element from St(n, R, I) ∩ K_{2}(n, R) belongs to St(n − 1, R). The usual arguments (see, for example [M]) show that any such an element is central. Thus F lies in the center of St(n, R). By proposition 1.3 St(n, R) is strongly polynormal in itself. Hence, F = [F, St(n, R)] = {e}, i.e. K_{2}(n, R) is central.

(5.5) Lastly, let us consider the following example. Let R be an arbitrary commutative ring and R⟨x, y⟩ be the free (associative) algebra over R. We consider the quotient A = R⟨x, y⟩/(xy = 1) of the algebra R⟨x, y⟩ modulo the relation xy = 1. Let us denote by I the smallest two-sided ideal in A containing the element yx − 1. It was shown in [St1] that sr(A, I) = 2. On the other hand, it is clear that the ring A/I is commutative. Hence, by W. van der Kallen’s theorem [vdK] K_{2}(n, A/I) is central if n ⩾ 4. Theorem 5.4 implies that K_{2}(n, A) is also central.

It follows from theorem 4.6 that the normal structure of GL(n, A) is standard if n ⩾ 3. The latter can be deduced also from the results of [Vs2] and [St1].

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