Elementary nets (carpets) over a discrete valuation ring Текст научной статьи по специальности «Математика»

УДК 512.5

Elementary nets (carpets) over a discrete valuation ring

Vladimir A. Koibaev*

North-Ossetian State University Vatutina, 44-46, Vladikavkaz, 362025 SMI VSC RAS Markusa, 22, Vladikavkaz, 362027

Russia

Received 24.06.2019, received in revised form 16.08.2019, accepted 20.09.2019 Elementary net (carpet) a = (ац) is called closed (admissible) if the elementary net (carpet) group E(a) does not contain a new elementary transvections. The work is related to the question of V. M. Levchuk 15.46 from the Kourovka notebook( closedness (admissibility) of the elementary net (carpet)over a field). Let R be a discrete valuation ring, K be the field of fractions of R, a = (ац) be an elementary net of order n over R, ш = (шц) be a derivative net for a, and шц is ideals of the ring R. It is proved that if K is a field of odd characteristic, then for the closedness (admissibility) of the net a, the closedness (admissibility) of each pair (ац ,aji) is sufficient for all i = j.

Keywords: nets, carpets, elementary net, closed net, derivative net, elementary net group, transvections,

discrete valuation ring.

DOI: 10.17516/1997-1397-2019-12-6-728-735.

Let K be a field, v a discrete valuation of the field K, R be discrete valuation ring, that is, R be the valuation ring of the v of the field K (field of fractions of the ring R). We consider an elementary net of order n (elementary carpet) a = (aij) additive subgroups of the ring R, associated with a derivative net ш = (w^). It is proved that if K is a field of odd characteristic and Uij are ideals of R, then for the closedness (admissibility) of the net a, the closedness of each pair (aij,aji) is sufficient for all i = j. In the considered case, a positive answer was received to the question of V. M. Levchuk (Kourovskaya notebook [1, question 15.46]) for a special linear group about reduction of the admissibility of an elementary carpet A = {Ar : r e Ф} to its admissibility subcarpets {Ar, A_r}, r e Ф of rank 1. In other words, for the case under consideration it was proved (Theorem 1) that the inclusion elementary transvection of tij (a) into an elementary group E(a) is equivalent to including tj(a) in the group (tj(aj),tji(aji)) (for any i = j). In fact, in theorem 1 it is proved (without restrictions on the characteristic of the field K), which of the inclusion tij(a) e E(a) follows inclusion tij(2a) e (tij(aij),tji(aji)). In the final part of the article, we look at an example a symmetric elementary net over the field of rational functions F(x) (over the field of coefficients F) and investigate it (Theorem 2) on closedness for an arbitrary field F other than the field F4 of four elements. Built examples (see Remark 1) show that the closure of the elementary net is arithmetic character, namely, it essentially depends on the characteristic of the field.

Note that the description of elementary (and complete) nets over locally finite field and the field of fractions of a principal ideal ring are obtained in [2, 3].

In the paper the following standard notations are adopted: R is an arbitrary commutative ring with a unit (in Sections 3 and 4 of R is discrete valuation ring); n is a natural number,

* koibaev-K1@yandex.ru © Siberian Federal University. All rights reserved

n ^ 3, a = (aij) is an elementary net over the ring R of order n. Let e is the identity matrix of order n, ej is the matrix, its entries at (i; j) are equal to 1, and all other entries are equal to zero; tij(a) — e + aeij is an elementary transvection. Let further tij(A) — {tij(a) : a G A}. For elementary net (carpet) a we consider the elementary net group E(a) and its subgroup

Eij(a), i = j:

E(a) = {tij(aij) :1 < i = j < n), Eij(a) = {tij(aij),tji(aji)).

Let F be a field, then through F(x) = j — : f,g G F[x], g = ^ denotes the field of rational functions with coefficients from F.

1. Discrete valuation rings

Let K be a field. A discrete valuation on K is a mapping v of the group K* = K\0 onto Z such that [ 4, ch. 9]

v(xy) = v(x) + v(y), v(x + y) > min(v(x), v(y))

(i.e., v is a surjective homomorphism). The set R consisting of 0 and all x G K* such that v(x) > 0 is a ring, called the valuation ring of v (it is a valuation ring of the field K). An integral domain R is a discrete valuation ring [4, ch. 9] if there is a discrete valuation v of its field of fractions K, such that R is the valuation ring. Let R be discrete valuation ring, then R is a local ring, and its maximal ideal m coincides with the set of those elements x G K, for which v (x) ^ 1: m = R\ = {x G R : v(x) ^ 1}, every ideal of R has the form Rn = {x G R : v(x) ^ n}, (n G N). Let 2 G R be an element such that v(z) = 1. Then Rn = (zn) = znR, in particular, m = R\ = zR. The group of invertible elements R* of the ring R coincides with the set {x G R : v(x) = 0} and 1+ m C R*.

Consider an example of a discrete valuation ring, which we will be used further in Section 4. Let F be a field, F(x) be a field of rational functions with coefficients from F. We define

the function v : F(x)\0 —> Z as follows. For — = 0 we set v(-) = deg(g) — deg(f). So the

g -

defined surjective function v satisfies the properties: v^f • = v(+ v(-2), v^f + >

min -1), v( —2) j. Therefore, K = F(x) is a discrete field of the valuation v, R = R0 = {- G F(x) : deg(g) > deg(f ^ is a discrete valuation ring. For non-negative integer m > 0 consider the set

'f- G F(x) : v(f-gg

Then every ideal of the discrete valuation ring R = R0 has kind Rm and is the main ideal Rm = — R = (Ri)m, next R is a local ring, m = R\ = — R is the maximal ideal of a local ring

Rm = { - G F(x) : v f = deg(g) — deg(f) > m}. (1)

R,

R* = { - G R : deg(g) = deg(f^, 1 + Rx C R*.

2. Preliminary results

System a = (aij), 1 < i,j < n, additive subgroups of a ring R are called net (carpet) [5, 6] over ring R of order n if airarj C aij for all values of the index i,r,j. A net viewed without a

diagonal is called elementary net (elementary carpet) [5-6, 1, question 15.46]. The elementary net a = (aij), 1 < i = j < n, is called supplemented if for some additive subgroups (more precisely, subrings) aii of the ring R table (with diagonal) a = (aij), 1 < i,j < n, is (full) net. It is well known (see, for example, [5]) that the elementary net a = (aij) is supplemented if and only if aijajiaij C aij for any i = j. Diagonal subgroups aii are defined by the formula

an = ^^ akiaik, (2)

k=i

where summation is taken over all k other than i.

A full or elementary net a = (aij) is called irreducible if all additive subgroups aij are different from zero. The elementary net a is called closed (admissible) if the subgroup E(a) does not contain new elemental transvection. Closed are, for example, elementary nets, the diagonal of which can be supplemented with subgroups, getting at the same time (full) net.

Let a = (aij) is an elementary net over the ring R of order n ^ 3. Consider the set w = (wij)

n

additive subgroups wij of ring R, defined for any i = j as follows: Wj = ^ aikakj, where

k=i

summation is taken over all k other than i and j. Set w = (wij) is a supplemented elementary net. We will add elementary net w to (full) net in a cyclical way, proposed in [7], setting wii = aikaksasi, where summation is done by all 1 < k = s < n. We call the constructed net

k=s

a derivative net (for elementary net a). Further, for arbitrary i = j we set Qij = aij + aijjj, where

Yij Qij Qji ^ ^ (ajiaij) , i = j.

m=i

The table Q = (Qij) is an elementary net, moreover supplemented, then there are fair inclusions QijQjiQij C Qij for any i = j. By virtue of (2) we add the elementary net Q to the (full) net,

putting Qii = Y^ QikQki, where the summation is taken by k, k = i. The net Q is called the

k=i

net associated with the elementary group E(a).

Lemma 1. For any pairwise distinct i,r,j there are inclusions: QirQrj C wij. Proof. Let i,r,j be pairwise distinct integers. In the beginning, we note that

air (ajr arj ) C air, arj (air ari) C arj

(the first inclusion is obvious, the second follows from the fact that arj(ariair) = ari(airarj) C arj). Therefore

(air + air (air ari) )(arj + arj (ajr arj ) ) air arj + air arj (ajr arj ) + (air arj )(air ari) (ajr arj )s + (a

ir arj )(air ari) — air arj air (arjajr ) \arj + air (arj ajr )s][( ariair) arj air [(ariair ) arj\ C airarj C wij C Qij .

□

Lemma 1 and Theorem 1 [7] imply the following proposition.

Proposition 1 ( [7], Theorem 1). Elementary net a induces a derived net w and the net Q, associated with the elementary group E(a), with w Ç a Ç Q, and for any i,r,j relations are fulfilled

{W ir QQ r j , QQ ir ^W r j ^W ij .

Further, for any pairwise different i,r,j there are inclusions: QirQrj Ç w^

Proposition 2 ( [8], Proposition 2). Let be a is an elementary net of order n over R, Q - a net associated with the elementary group E(a). If a = (5j + aj) £ E (a), then a^ £ Qj.

Using the nets w = (wj) and Q = (Qj), which are defined for the elementary net a, we will build a new net t follows. In the elementary net Q to the position (1, 2) instead of Q12 we put w12, and the position (2, 1) instead of Q21 we set w21. According to proposition 1 table thus obtained will be an elementary net, and it is a supplemented elementary net. We will add its up to the (full) net as follows: we set th — ^Qn, i = 3, ... ,n,

t11 = W11 + Q13Q31 + • • • + QlnQnl, t22 = w22 + Q23Q32 + • • • + Q2nQn2-

According to proposition 1, the table t is a net and has the form:

/tii Wi2 fiis . . fi in ^

W2i T22 ^23 . .. fi2n

T= fisi fis2 fiss . . fisn

\fini fin2 finS . .. finn

Proposition 3 ([9], Theorem). Let n > 3, t2i(a) G E(a). Then t2i(a) = ah, a G Ei2(a), h G G(t). If

a = diag

. f fl + an ai2 A \ u A- ( (1 + hii h 12 A A

iagU a2i 1 + 0*2) >en-2J, h = diag[{ h2i 1 + h22) >en-2J,

( 1 0\ = (1 + aii ai2 \(1 + hii h 12 N \a 1) V a2i 1 + a22j\ h2i 1 + h22j '

(3)

then au, hu G Tii n T22 n 7i2- i = 1, 2,

aii = h22, a, a2i G fi2i, a - a2i G W2i, ai2, hi2 G Wi2, h2i G W2i,

a22 + aii, a22 - hii, a22 + h22, hii + h-22, hii + aii G un n u22.

Proposition 4. Let the conditions of proposition 3 be satisfied. Then

(1) aiia2i G U2i;

(2) if t2i(2a2i) G Ei2(a), then t2i(2a) G E^(a).

Proof. (1) By the condition of proposition 3 a2i G fi2i, aii = h22 G t22, but then according to Lemma 1 [9] we have T22fi2i Ç w2i, whence aiia2i G w2i. (2) According to proposition 3, we have

a - a2i G W2i Ç <T2i t2i(2(a - a2i)) G Eu(a) =^ t2i(2a) G Ei2(a).

□

3. The main result

Let K be a field, v a discrete valuation of the field K. In this section R is a discrete valuation ring, that is R = {x G K : v(x) ^ 0} - valuation ring of the v of the field K (fields of fractions of the ring R).

Theorem 1. Let a = (aij) - elementary net over a discrete valuated ring R, u = (uij ) is a derivative net, and uij is the ideal of R for all i = j. If tij (a) G E(a), then tij (2a) G Ej (a) = = {tij (aij ),tji(aji)). In particular, if K is afield of odd characteristic, then tij (a) G Eij (a).

Proof. Without loss of generality, we set i = 2, j = 1. Let m be the maximal ideal of R. If w21 = R (and then a21 = R), then the conclusion of the theorem obviously, so we will assume that w21 Ç m. Let t21(a) G E(a). Then, according to proposition 3, we have t2i(a) = ah, a G Ei2(a), h G G(t),

A- ( (l + aii a 12 ) ) , ( (1 + hii hi2 ) )

a = diai{ a2i 1 + a22j >en-2) , h = diai{ h2i 1 + h22) ,en-2) ,

and equality (3) holds. To prove the theorem according to Proposition 4 (2), it suffices to show that t21(2a21) G E12(a) = (t12(a12),t21 (a21)). From (3), according to Proposition 3, we have (ai2h21 G W21 ■ W12)

(1 + an)(1 + hn) G 1 + W21 ■ W12 C 1 + m C R* (1 + an)(1 + hn) G R*.

Therefore

0 = v[(1 + au)(1 + hn)] = v(1 + an) + v(1 + hn) =^ v(1 + an) = 0.

Consequently, 1 + a11 is an invertible element of the ring R. Further, because a12 G w12 h w12 is

an ideal of ring R, then -—G w12 C a12. Therefore, the inclusion a G E12(a) implies what 1 + an

at 12 (—^) = (1 + a11 (1+ ° -1) G E12(a). (4)

\1 + auJ V a21 (1 + an)

Further, since ( ( )T is the transposition of the matrix and (12) is matrix-permutation)

"2)(a <)(a <) (12)(1 <1)t<12)=(11).

then the matrix

(12) (1 + an 0 )T (12)= ((1 + an)-1 0 )

( )V a21 (1 + an)-1) (12)={ a21 (1 + an))

also is contained in the group E12(a). Here it should be noted that the matrix (4) is represented as a product of elementary transvections from t21(a21) and t12(a12). It follows that the matrix

1 0\ = /(1 + aii)-1 0 \(1 + aii 0

V V a2i (1 + aii))\

2a21(1 + aii) 1j \ a2i (1 + aii)J \ a2i (1 + aii)

i

is contained in E12(a). According to proposition 4 (1) a21a11 G w21 Ç a21, hence t21(a21a11 ) G

E12(a), and therefore t21 (2a21 ) G E12(a). Then, according to proposition 4 (2) t21(2a) G E12(a).

□

4. Examples of not closed symmetric nets over a field of rational functions

Let F be a field, F(x) be a field of rational functions with coefficients from F. Consider the discrete valuation v of the field F(x) (see section 1): v(= deg(g) — deg(f ). Then R = R0 =

(

= | — € F(x) : deg(g) > deg(f ^ is the valuation ring of v. The ideals of the ring R have the

1

form Rm, m ^ 0 (see (1)). Further, m = R1 = — R is the maximal ideal of a local ring R.

x

F

Put B =--+ R4. Consider the table t = (tj) of order n > 2, for which t12 = t21 = B,

Tij = R4 for the remaining i = j:

* B R4 . .. R4

B * R4 . .. R4

T = R4 R4 R4 . .. R4

R4 R4 R4 . .. *

The table t is an elementary net since R4R4 C R4 C B, R4B C R4, and t is not supplemented elementary net as 3 £ B3\B.

F ( * B\ Theorem 2. Let B =--+ R4. If |F| ^ 5, then elementary net a = ( b * J is not closed (in

particular, t is not closed).

Proof. Put z = 1 £ B, B = Fz + R4. Then for any £ £ F, £ = 0 an element £z3 =

x £x3

is not contained in the subgroup B. For the proofs of the theorem are sufficient to show that tl2(£z3) is contained in E(a) = (t12(B),t2l(B)) for some £ £ F, £ = 0. Now the proof of the theorem follows from the following Lemma 2. □

Lemma 2. Let IF| > 5. Put q £ F, q = 0, q = 1, q2 - q + 1 = 0. Set

b = q2 - q +1 £ F, zi = -q2z\ - qzZ £ Re, 1 + zi £ 1 + R6 £ R*.

b

Then

[t2l(z),ti2(z)] ■ t2i (qz)ti^ —-1) t2i( z(q - 1)2) ti2 ( ^^ )

Xt2l(

(z),ti2(z , „(„ _ 1

(q - 1)b;

-qz + q(q - 1)z7) (q2z + qz

1 + zi

>12 ( yn(-bz>

23

-qz + q2 z3

£ £ E(a),

ti2l ' i '-+ £) £ E(a)

-qz _ n _______-qz , ^ ^ r, j-L___ -t (q-z~

where £ £ R5 C B, —— £ B. In particular, since —--+ £ £ B, then tl2(^b^) £ E(a).

b b 0

Proof. We set

S = [t2l(z),ti2(z)] ■ t2i(qz)ti2 (—^ t2l (z(q - 1)2)tl2 ( q - ^ )

-qz + q2z3 - q(q - 1)z5 - qz7 ]'

It is easy to check the formula (

S=

[1 - qz2 + q2z - q(q - 1)z6]

^ [bz + qz5 - q(q - 1)z7]

1

q2z6 + qz8 ]

b

Note that (S)22 = 1 + zl £ R*. Recall that zl = q z ,——. Therefore

b

St -qz5 + q(q - 1)z7 )= I [1 - qz2 + q2z4 + £i] 2l( 1 + zi ) I bz

-qz + q2z3 - q(q - 1)z5 - qz7

b

1 + zi

x

where ^ £ R6. Therefore

-qz5 + q(q - 1)z7 q2z5 + qz7

S • f2l(--)t12(-v2-)

1 + z1 b2

2 3

[1 - qz2 + q2z4 + £1] IZl^+l^ + £

bz

where £ £ R5 C B. Consequently,

f-qz5 + q(q - 1)z7 \ f q2z5 + qz7 \ f -qz + q2z3 A

S • M-1T7i-)tl2 (—-)t2l(-bz) = tl2{-b-+ V ■

□

Remark 1. If F = F2; then the elementary net t is closed (see [10]). Theorem 2 is also valid for the case of a F = F3 field of three elements (we do not provide evidence because of its bulkiness). Thus, for a full study on the closure of the elementary net t remains to consider the case of a field F = F4 of four elements.

Remark 2. Example of the elementary net t allows you to build similar examples over arbitrary a field in which there is a transcendent element over the simple subfield of K.

Remark 3. An example of the elementary net t allows you to build similar examples over arbitrary a discrete valuation ring R (valuation ring of the discrete valuation field of v (see section 1)), which in addition is a F-algebra over the field F: set B = Fz + z4R, where

v(z) = 1, m = (z) = zR, R = {x £ K : v(x) > 0}, Fz3 n B = 0.

For example, if ^ is an irreducible polynomial of the ring F[x] and the valuation of v in the field

of rational functions F(x) is determined according to the degree of a polynomial ^ in a rational f

fraction —, (f,g) = 1. Then R = F(x)iv\ coincides with the localization of the polynomial ring

g

F[x] with respect to a prime ideal (<p), further, m = (^) = y • R, B = F • y + y.>4 • R.

References

[1] The Kourovka notebook : unsolved problems in group theory, Russ. acad. of sciences, Siberian div., Inst. of mathematics. Novosibirsk, Vol. 17, 2010 (in Russian).

[2] V.M.Levchuk, Generating sets of root elements of Chevalley groups over a field, Algebra and Logic, 22(1983), no. 5, 504-517.

[3] R.Y.Dryaeva, V.A.Koibaev, Ya.N.Nuzhin, Full and elementary nets over the field of fractions of a principal ideal ring, Journal of Mathematical Sciences, 234(2018), no. 2, 141-147.

[4] M.F.Atiyah, I.G.Macdonald, Introduction to commutative algebra, Addison-Wesley, 1969.

[5] Z.I.Borevich, Subgroups of linear groups rich in transvections, Journal of Soviet Mathematics, 37(1987), no. 2, 928-934.

[6] V.M.Levchuk, A Note to L.Dickson's Theorem, Algebra and Logic, 22(1983), no. 4, 421-434.

1

[7] N.A.Dzhusoeva, S.Y.Itarova, V.A.Koibaev, On the embedding an elementary net into a gap of nets, XVI International Conference Algebra, Number Theory and Discrete Geometry: modern problems, applications and problems of history, Tula. Abstracts, 73-74.

[8] R.Y.Dryaeva, V.A.Koibaev, Decomposition of elementary transvection in elementary group, Journal of Mathematical Sciences, 219(2016), no. 4, 513-518.

[9] S.Y.Itarova, V.A.Koibaev, Decomposition of elementary transvection in elementary net group, Vladikavkaz. Mat. Zh., 21(2019), no. 3.

[10] V.A.Koibaev, Elementary nets in linear groups, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 17(2011), no. 4, 134-141.

Элементарные сети (ковры) над дискретно нормированным кольцом

Владимир А. Койбаев

Северо-Осетинский государственный университет Ватутина, 44-46, Владикавказ, 362025 ЮМИ ВНЦ РАН Маркуса, 22, Владикавказ, 362027

Россия

Элементарная сеть (ковер) а = (aij) называется замкнутой(допустимой), если элементарная сетевая (ковровая) группа Е(а) не содержит новых элементарных трансвекций. Работа связана с вопросом В. М. Левчука 15.46 из Коуровской тетради о замкнутости (допустимости) элементарной сети (ковра) над полем. Пусть R — дискретно нормированное кольцо, K — поле частных кольца R, а = (aij) — элементарная сеть (ковер) порядка n над R, ш = (ш^) - производная сеть для а, причем — идеалы кольца R. Доказано, что если K — поле нечетной характеристики, то для замкнутости (допустимости) сети а достаточна замкнутость (допустимость) каждой пары (aij ,aji) для всех i = j.

Ключевые слова: сети, ковры, элементарная сеть, замкнутая сеть, производная сеть, элементарная сетевая группа, трансвекция, дискретно нормированное кольцо.