MATHEMATICS

M. Sh. KHUZURBAZAR

ON THE THEORY OF MULTIPLICATIVE GROUPS OF DIVISION RINGS

(Presented by Academician P. S. Aleksandrov, 22 X 1960)

§ 1. Let \(K\) be an associative noncommutative division ring, and let \(Z\) be the center of the division ring \(K\). The set of all nonzero elements of the division ring \(K\) forms a group under multiplication, which we shall denote by \(K^*\). If \(Z^*\) is the set of all nonzero elements of \(Z\), then \(Z^*\) is the center of the group \(K^*\). In \((^1)\) the author proved that every locally nilpotent normal divisor of the group \(K^*\) is contained in the center \(Z^*\), and, consequently, the factor group \(K^*/Z^*\) has no nontrivial locally nilpotent normal divisors. In particular, \(K^*\) is not an \(RN^*\)-solvable group, i.e. \(K^*\) does not possess an ascending solvable normal series. Indeed, every \(RN^*\)-solvable normal divisor of the group \(K^*\) is contained in the center \(Z^*\).

In the present note it is proved that these results are in a certain sense final. The following general theorem holds:

Theorem 1. Every division ring \(K\) can be embedded in a division ring \(D\), whose multiplicative group \(D^*\) possesses an infinite descending invariant series

\[ D^* \supset A_1 \supset A_2 \supset A_3 \supset \cdots,\qquad \bigcap_{j=1}^{\infty} A_j=C^*, \]

where: 1) \(C\) is the center of the division ring \(D\); 2) \([A_1,A_j]\subseteq A_{j+1}\), \(j=1,2,\ldots\), where \([A_1,A_j]\) is the mutual commutant of the subgroups \(A_1\) and \(A_j\); 3) \(D^*/A_1\) is isomorphic to \(K^*/Z^*\).

Proof. Denote by \(D\) the ring of formal power series of the form \(\sum_{i\ge m} a_i t^i\), where \(a_i\in K\), \(m\) is an integer, and the indeterminate \(t\) commutes with all elements of the division ring \(K\). Then \(D\) is a division ring, as a special case of the Hilbert division rings defined in \((^2)\) (pp. 187–188). The center \(C\) of the division ring \(D\) consists of all series of the form \(\sum_{i\ge m} z_i t^i\), where \(z_i\in Z\). It is clear that \(D\) contains \(K\).

We now show that the division ring \(D\) has all the properties formulated in the theorem. If \(a=\sum_{i\ge m} a_i t^i\) and \(a_m\ne 0\), then we shall agree to call \(a_m\) the first coefficient of the element \(a\), and the coefficients \(a_{m+1}, a_{m+2},\ldots\), respectively, the second, third, etc.

Let \(A_j\) be the set of all elements of \(D^*\) whose first \(j\) coefficients belong to \(Z\). We shall prove that each \(A_j\) is a normal divisor of the group \(D^*\). Indeed, let \(\alpha,\beta\in A_j\). It is easy to see that the element \(\alpha\beta\) also belongs to \(A_j\). If \(\alpha\in A_j\), then \(\alpha^{-1}\) also belongs to \(A_j\), as one verifies by comparing the first \(j\) coefficients in the equality \(\alpha\alpha^{-1}=1\) and taking into account that the first \(j\) coefficients of the element \(\alpha\) belong to the center \(Z\). Now let \(\alpha\in A_j\) and let \(\beta\) be an arbitrary element of the group \(D^*\). Then the first \(j\) coefficients of the element \(\gamma=\beta\alpha\beta^{-1}\) coincide with the corresponding coefficients of the element \(\alpha\), as we verify by comparing the first \(j\) coefficients in the equality \(\gamma\beta=\beta\alpha\). Consequently, \(\gamma\) also belongs to \(A_j\). Thus, indeed, \(A_j\) is a normal divisor of the group \(D^*\).

1. From the definition of the sets \(A_j\) it follows that

\[ A_1 \subset A_2 \subset A_3 \subset \cdots,\qquad \bigcap_{j=1}^{\infty} A_j=C^*, \]

where \(C\) is the center of the division ring \(D\).

1. We shall prove that \([A_1,A_j]\subseteq A_{j+1}\). This means that for any \(a\in A_j\), \(\beta\in A_1\), the element \(\beta a\beta^{-1}a^{-1}\in A_{j+1}\). As was already mentioned, for \(a\in A_j\) and any \(\beta\in D^*\), the first \(j\) coefficients of the element \(\beta a\beta^{-1}\) coincide with the corresponding coefficients of the element \(a\). If, moreover, \(\beta\in A_1\), then the corresponding first \(j+1\) coefficients also coincide. Therefore \((\beta a\beta^{-1})a^{-1}\) has the form \(1+b_{j+1}t^{j+1}+\cdots\), i.e. \(\beta a\beta^{-1}a^{-1}\in A_{j+1}\), as was required to prove.

2. To complete the proof of the theorem it remains to show that \(D^*/A_1\) is isomorphic to \(K^*/Z^*\). Define a mapping \(\varphi:D^*\to K^*\) as follows. If \(a\) is an arbitrary element of \(D^*\) and \(a_m\) is the first coefficient of the element \(a\), then \(a\varphi=a_m\). It is easy to see that the mapping \(\varphi\) is a homomorphism of the group \(D^*\) onto the group \(K^*\). The kernel \(N\) of this homomorphism consists of all elements of \(D^*\) whose first coefficient is equal to 1. Thus \(N\subseteq A_1\). Consequently, the homomorphism \(\varphi\) induces a homomorphism of the group \(A_1\) onto \(Z^*\) with the same kernel \(N\). Hence \(D^*/N\simeq K^*\), \(A_1/N\simeq Z^*\). From this,
   \[ (D^*/N)/(A_1/N)\simeq K^*/Z^*, \]
   i.e. \(D^*/A_1\simeq K^*/Z^*\). The theorem is proved.

Remark 1. Each normal divisor \(A_j\), \(j=1,2,\ldots\), of the group \(D^*\) is a \(ZD\)-group, i.e. \(A_j\) has a descending central series. In particular, \(A_j\) is an \(RK\)-group and, still more, an \(RN\)-group, i.e. \(A_j\) has a solvable normal system. However, \(A_j\) is not contained in the center \(C^*\). It is clear that in the factor group \(D^*/C^*\) there exist nontrivial normal divisors \(A_j/C^*\), which are \(RN\)-groups \((RK\)-groups, \(ZD\)-groups).

Remark 2. From the primarity of the factor group \(K^*/Z^*\) (see \((^1)\)) it follows that \(A_1\) is a primary normal divisor of the group \(D^*\). Thus, for the multiplicative group of a division ring, the center need not be the only primary normal divisor.

Remark 3. In \((^2)\), p. 191, an example is given showing that the multiplicative group of a division ring may contain an infinite descending invariant series. Our theorem shows that such a situation is quite general.

§2. The Cartan–Brauer–Hua theorem (see \((^2)\), p. 186) establishes that if \(M\) is a proper subdivision ring of a division ring \(K\), not contained in the center \(Z\) of the division ring \(K\), then the multiplicative group \(M^*\) is not a normal divisor of the group \(K^*\). In \((^{3,4})\) it is shown that in this case the group \(M^*\) is not even a member of any finite normal series of the group \(K^*\). Here we give a further generalization of this result.

Theorem 2. Let \(M\) be a proper subdivision ring of a division ring \(K\), not contained in the center \(Z\) of the division ring \(K\). Then the multiplicative group \(M^*\) is not a member of any ascending (in general, transfinite) normal series of the group \(K^*\).

As usual, we shall call a subgroup \(H\) subinvariant in a group \(G\) if \(H\) is a member of some ascending (in general, transfinite) normal series of the group \(G\). B. I. Plotkin \((^5)\) proved that if \(R(G)\) is the radical of the group \(G\), then for any subinvariant subgroup \(H\) of the group \(G\) the relation \(R(H)=R(G)\cap H\) holds. On the other hand, the author showed \((^1)\) that for the multiplicative group \(K^*\) of the division ring \(K\) the radical coincides with the center \(Z^*\). Consequently, if \(M^*\) is subinvariant in \(K^*\), then the center of the division ring \(M\) must lie in the center of the division ring \(K\). Therefore, if \(M\) is in fact a field not contained in \(Z\), then \(M^*\) cannot be a subinvariant subgroup of the group \(K^*\). Thus the theorem is proved for the case when the subdivision ring \(M\) is commutative.

To prove the theorem in the general case, we shall need some lemmas. Denote
\(V(M)=\{x\in K\mid xm=mx \text{ for all } m\in M\}\) and
\(N(M)=\{y\in K\mid y^{-1}My=M\}\). \(V(M)\) is called the centralizer of the subfield \(M\) in \(K\).

Lemma 1 (see (²), p. 186). Let \(L\) and \(M\) be subfields of the field \(K\) and let \(L^*\subseteq N(M)\). Then either \(L\subseteq M\), or \(L\subseteq V(M)\).

Lemma 2. Let \(M\) be a noncommutative proper subfield of the field \(K\), and let \(V(M)\) be commutative. Then \(M^*\) is not a subinvariant subgroup in \(K^*\).

Proof. Suppose that \(M^*\) is subinvariant and
\(M^*\subset A_1\subset A_2\subset\cdots\subset A_\alpha\subset A_{\alpha+1}\subset\cdots\subset A_\gamma=K^*\) is an ascending normal series passing through \(M^*\). We shall show that every term of this series is contained in \(N(M)\). This will mean that also \(A_\gamma=K^*\subseteq N(M)\), i.e. \(M^*\) is a normal divisor of the group \(K^*\), which contradicts the Cartan–Brauer–Hua theorem.

By construction, \(A_1\subseteq N(M)\). Suppose that for all \(\beta<\alpha\), \(A_\beta\subseteq N(M)\). If \(\alpha\) is a limit transfinite number, then
\[ A_\alpha=\bigcup_{\beta<\alpha} A_\beta\subseteq N(M). \]
If, however, \(\alpha-1\) exists, then for any \(x\in A_\alpha\) we have
\[ x^{-1}M^*x\subseteq x^{-1}A_{\alpha-1}x=A_{\alpha-1}\subseteq N(M), \]
i.e. \((x^{-1}Mx)^*\subseteq N(M)\). In view of Lemma 1, either \(x^{-1}Mx\subseteq M\), or \(x^{-1}Mx\subseteq V(M)\). Since \(V(M)\) is commutative and \(M\) is noncommutative, \(x^{-1}Mx\not\subset V(M)\). Consequently, \(x^{-1}Mx\subseteq M\) for any \(x\in A_\alpha\). Since \(A_\alpha\) is a group, \(x^{-1}Mx=M\) for any \(x\in A_\alpha\), i.e. \(A_\alpha\subseteq N(M)\). The lemma is proved.

Let us note that if \(V(M)\subset M\), then \(V(M)\) is commutative. Thus, we have:

Corollary. Let \(M\) be a noncommutative proper subfield of the field \(K\) and let \(V(M)\subset M\). Then \(M^*\) is not a subinvariant subgroup in \(K^*\).

We formulate one more lemma, which can be verified.

Lemma 3. Let \(M\) be a subfield of the field \(K\) and suppose that there exists an element \(a\in V(M)\), \(a\notin M\). Let \(\Delta\) be the minimal subfield containing \(a\) and \(M\). Then the centralizer of the subfield \(M\) in \(\Delta\) coincides with the center of the field \(\Delta\) and, consequently, is commutative.

Proof of Theorem 2. We have already proved the theorem for the case when \(M\) is commutative. Let \(M\) be a noncommutative proper subfield of the field \(K\). Suppose that \(M\) is subinvariant in \(K^*\). By the corollary to Lemma 2, \(V(M)\) is not contained in \(M\). Let \(a\in V(M)\), \(a\notin M\), and let \(\Delta\) be the minimal subfield of the field \(K\) containing \(M\) and \(a\). Then \(M\) is a proper subfield of the field \(\Delta\), and, by Lemma 3, the centralizer of the subfield \(M\) in \(\Delta\) is commutative. By Lemma 2, \(M^*\) is not a subinvariant subgroup in the group \(\Delta^*\). But \(M^*\), being subinvariant in \(K^*\), must be subinvariant in every subgroup of the group \(K^*\) containing \(M^*\). This contradiction proves the theorem.

I express my deep gratitude to Prof. A. G. Kurosh for supervising the work.

Moscow State University
named after M. V. Lomonosov

[Received
20 X 1960]

References

¹ M. Sh. Khuzurbazar, DAN, 131, No. 6 (1960).
² N. Jacobson, Am. Math. Soc. Coll. Publ., 37 (1956).
³ E. Schenkman, Proc. Am. Math. Soc., 9, No. 2 (1958).
⁴ E. Schenkman, W. R. Scott, Proc. Am. Math. Soc., 11, No. 3 (1960).
⁵ B. I. Plotkin, Tr. Mosk. matem. obshch., 6, 299 (1957).