A. A. BOVDI

ON THE EMBEDDING OF CROSSED PRODUCTS IN DIVISION RINGS

(Presented by Academician P. S. Aleksandrov, 4 III 1963)

In this note we generalize the well-known Mal'cev–Neumann theorem on the embedding of group algebras of ordered groups in division rings, as well as Ikeda’s result \((^1)\) stating that the crossed product \((G,D,\rho,\sigma)\) of an \(RN^*\)-group \(G\) with factors without twisting and a division ring \(D\), where \(\sigma\) is a mapping with trivial kernel, has a division ring of right quotients. (For the definition of a crossed product and the necessary notation, see \((^3)\).)

Lemma 1. Let \((G,D,\rho,\sigma)\) be the crossed product of a group \(G\) and a ring \(D\); let \(H\) be an invariant subgroup of the group \(G\). If the subring \((H,D,\rho,\sigma)\) has a division ring of right quotients \(T\), then the mapping
\[ xy^{-1}\to t_g^{-1}xy^{-1}t_g=t_g^{-1}xt_g\,(t_gyt_g^{-1})^{-1}, \]
where \(x,y\in (H,D,\rho,\sigma)\), \(g\in G\), is an automorphism of the division ring \(T\).

The assertion of the lemma follows immediately from the fact that the mapping \(x\to t_g^{-1}xt_g\) is an automorphism of the ring \((H,D,\rho,\sigma)\).

Lemma 2. Let \((G,D,\rho,\sigma)\) be the crossed product of a group \(G\) and a ring \(D\), containing no zero divisors, and let \(H\) be an invariant subgroup of the group \(G\) such that \(G/H\) is an abelian group with a finite number of generators. If there exists a ring of right quotients \(L\) of the ring \((G,D,\rho,\sigma)\) relative to the subring \((H,D,\rho,\sigma)\), then the ring \(L\) is Noetherian.

Proof. Let
\[ G/H=\{\bar d_1\}\times\cdots\times\{\bar d_t\}\times \bar H_1 \]
and
\[ \bar K=\{\bar d_1\}\times\cdots\times\{\bar d_{t-1}\}\times \bar H_1, \]
where \(\bar H_1\) is a finite abelian group, \(\{\bar d_i\}\) is an infinite cyclic group, \(i=1,\ldots,t\); \(K(H_1)\) is the full preimage of the group \(\bar K(\bar H_1)\) in the group \(G\), and \(d^n\) is one of the preimages of the element \(\bar d_t^n\) in the group \(G\), \(n=\ldots,-2,-1,0,1,2,\ldots\). Every element of the ring of right quotients \(L_1\) of the ring \((H_1,D,\rho,\sigma)\) relative to the subring \((H,D,\rho,\sigma)\) can be represented in the form \(\sum t_{k_i}xy^{-1}\), where \(k_i\) are representatives of the cosets of the group \(H_1\) modulo the subgroup \(H\), \(x_i,y\in (H,D,\rho,\sigma)\). Since \(L_1\) is a finite-dimensional vector space over the division ring of right quotients of the ring \((H,D,\rho,\sigma)\), it follows that \(L_1\) is Noetherian. Suppose that the ring of right quotients \(L_2\) of the ring \((K,D,\rho,\sigma)\) relative to the subring \((H,D,\rho,\sigma)\) is Noetherian, and let us prove that \(L\) is Noetherian.

Let \(I\) be an arbitrary right ideal of the ring \(L\). Then, by Lemma 1, every element of \(I\) can be represented in the form
\[ x=\sum_{r\le n\le s} c_n t_{d^n}, \]
where \(c_n\in L_2,\ c_s\ne 0\); the element \(c_s\) will be called the leading coefficient of the element \(x\). The set of leading coefficients of the elements belonging to \(I\) forms a right ideal \(I_0\) of the ring \(L_2\). Indeed, if \(b_t\) is the leading coefficient of the element \(y\), then \(x\pm yt_h\rho_{d^t,h}^{-1}\), where \(h\) is determined from the equality \(d^t h=d^s\), has leading coefficient \(c_s\pm b_t\), and if \(w\in L_2\), then
\[ xt_{d^s}^{-1}wt_{d^s} \]
has leading coefficient \(c_sw\). Since \(L_2\) is Noetherian, \(I_0\) has a finite basis \(a_1,\ldots,a_l\). Let \(x_i\) be an element of the right ideal \(I\) with leading coefficient \(a_i\), and we may suppose that in the expression of the element \(x_i\) only \(t_{d^n}\), \(0\le n\), occur, and that \(x_i\) has leading term \(a_it_{d^m}\) (\(m\) is the same for all \(i\)).

Denote by \(I_1\) the right ideal of the ring \(L\) generated by the elements \(x_1,\ldots,x_l\). If \(x\) is an arbitrary element of the ideal \(I\), then there exists an element \(d^k\) such that, in the expression of the element \(y=xt_{d^k}\), only the \(t_{d^n}\), \(n\geq 0\), occur; and suppose that \(\widetilde a_p\) is the leading coefficient of the element \(y\). Then \(\widetilde a_p=\sum a_i y_i\), where \(y_i\in L_2\), and if \(p\geq m\), then
\[ y-\sum x_i t_{d^m}^{-1} y_i t_{d^p} \]
is an element of the right ideal \(I\), in whose expression only the basis elements \(t_{d^n}\), \(0\leq n\leq p-1\), occur. Repeating this argument, we obtain that \(y\equiv v\pmod{I_1}\), and in the expression of \(v\) only \(t_{d^n}\), \(0\leq n\leq m-1\), occur. The set \(I_2\) of elements of the ideal \(I\) which are linear combinations of the elements \(t_{d^n}\), \(0\leq n\leq m-1\), forms a right \(L_2\)-module. This module has a finite \(L_2\)-basis \(x_{l+1},\ldots,x_r\), since the set of leading coefficients is again a right ideal of the ring \(L_2\), and the preceding arguments may be repeated. If \(I_3\) is the right ideal of the ring \(L\) generated by \(x_1,\ldots,x_r\), then \(y t_{d^k}^{-1}=x\in I_3\). Consequently, \(I_3=I\). The lemma is proved.

Theorem. Suppose that the group \(G\) has a normal divisor \(H\) such that \(G/H\) is an ordered group, and \(H\) has an ascending normal series whose factors are locally finite over their centers. If \((G,D,\rho,\sigma)\) is an arbitrary crossed product of the group \(G\) and the skew field \(D\), and the subring \((H,D,\rho,\sigma)\) contains no zero divisors, then \((G,D,\rho,\sigma)\) can be embedded in a skew field, and the subring \((H,D,\rho,\sigma)\) has a skew field of right fractions.

Proof. Let
\[ 1=H_0\subset H_1\subset \cdots \subset H_\tau=H \]
be an ascending normal series of the group \(H\), with the factors \(H_{i+1}/H_i\) locally finite groups over their centers \(Z_{i+1}/H_i\), \(0\leq i<\tau\). We shall show that the ring \((H,D,\rho,\sigma)\) has a skew field of right fractions. Since for the ring \((H_0,D,\rho,\sigma)\) the induction hypothesis is satisfied, we may assume that the ring \((H_\gamma,D,\rho,\sigma)\) has a skew field of right fractions \(L_\gamma\) for all \(\gamma<\alpha\), and these skew fields are embedded in one another.

If \(\alpha\) is a limit ordinal, then the set-theoretic union of all \(L_\gamma\), \(\gamma<\alpha\), is a skew field of right fractions \(L_\alpha\) of the ring \((H_\alpha,D,\rho,\sigma)\). If there exists \(\alpha-1\), then there exists a skew field of right fractions \(L_{\alpha-1}\) of the ring \((H_{\alpha-1},D,\rho,\sigma)\).

Consider the set \(S\) of all finite sums
\[ \sum t_{k_i}x_i y_i^{-1}, \]
where \(x_i,y_i\in (H_{\alpha-1},D,\rho,\sigma)\), \(k_i\in \Pi(Z_\alpha/H_{\alpha-1})\) are representatives of the cosets of the group \(Z_\alpha\) by the subgroup \(H_{\alpha-1}\). If \(x,y,v,w\in(H_{\alpha-1},D,\rho,\sigma)\), \(k,k_1\in \Pi(Z_\alpha/H_{\alpha-1})\), and
\[ t_{k_1}^{-1} v t_{k_1}\,(t_{k_1}^{-1} w t_{k_1})^{-1}xy^{-1}=x_1y_1^{-1} \]
in the skew field \(L_{\alpha-1}\), then, by Lemma 1, multiplication may be defined in the set \(S\) as follows:
\[ t_k v w^{-1}t_{k_1}xy^{-1} = t_{k_2}\bigl(t_h\rho_{k_2,h}^{-1}\rho_{k,k_1}\,x_1y_1^{-1}\bigr), \]
where \(kk_1=k_2h\), \(k_2\in \Pi(Z_\alpha/H_{\alpha-1})\), and \(h\in H_{\alpha-1}\). Then, if we suppose that
\[ \sum t_{k_i}x_i y_i^{-1}=\sum t_{k_i}v_i w_i^{-1}, \]
then this holds if and only if \(x_i y_i^{-1}=v_i w_i^{-1}\) for all \(i\); with respect to the natural coordinatewise addition and the multiplication introduced, \(S\) will be a ring. Every element of the ring \((Z_\alpha,D,\rho,\sigma)\) can be represented in the form \(\sum t_{k_i}x_i\), where \(x_i\in(H_{\alpha-1},D,\rho,\sigma)\), \(k_i\in \Pi(Z_\alpha/H_{\alpha-1})\), and \((Z_\alpha,D,\rho,\sigma)\) will be a subring of \(S\). Further, since in the skew field \(L_{\alpha-1}\) one can reduce to a common denominator, \(S\) is the ring of right fractions of the ring \((Z,D,\rho,\sigma)\) with respect to the subring \((H_{\alpha-1},D,\rho,\sigma)\), and therefore it contains no zero divisors. We shall show that the ring \(S\) satisfies Ore’s condition. Let \(xy^{-1}, vw^{-1}\in S\) and
\[ x=\sum t_{k_i}y_i,\qquad v=\sum t_{l_i}w_i, \]
where \(y,y_i,w,w_i\in(H_{\alpha-1},D,\rho,\sigma)\), \(k_i,l_i\in\Pi(Z_\alpha/H_{\alpha-1})\). Denote by \(K\) the subgroup of the group \(H_\alpha\) generated by the elements \(k_1,\ldots,k_n,l_1,\ldots,l_m\) and by the group \(H_{\alpha-1}\). Then \(K/H_{\alpha-1}\) is an abelian group with a finite number of generators, and \(xy^{-1}, vw^{-1}\) are elements of the skew field of right fractions \(S_1\) of the ring \((K,D,\rho,\sigma)\) with respect to—

relative to the subring \((H_{\alpha-1}, D, \rho, \sigma)\). By Lemma 2, the ring \(S_1\) is nonzero, and in such rings, as Goldie showed (see (4)), the Ore condition is satisfied. Consequently, the ring \((Z_{\alpha}, D, \rho, \sigma)\) has a skew field of right quotients \(S_2\).

Similarly one can show the existence of a ring of right quotients \(S_3\) of the ring \((H_{\alpha}, D, \rho, \sigma)\) with respect to the subring \((Z_{\alpha}, D, \rho, \sigma)\). The ring \(S_3\) contains no zero divisors and is a skew field. Indeed, if

\[ x=\sum_{i=1}^{n} t_{k_i}x_i y_i^{-1}, \]

where \(x_i y_i^{-1}\in S_2,\ k_i\in \Pi(H_{\alpha}/Z_{\alpha})\), and \(M\) is the subgroup of the group \(H_{\alpha}\) generated by \(Z_{\alpha}\) and the elements \(k_1,\ldots,k_n\), then the factor group \(M/Z_{\alpha}\) is finite and \(x\) is an element of the ring of right quotients \(S_4\) of the ring \((M,D,\rho,\sigma)\) with respect to the subring \((Z_{\alpha},D,\rho,\sigma)\). The ring \(S_4\) is a finite-dimensional vector space over a skew field and has no zero divisors. Hence \(x\) is an invertible element and \(S_3\) is a skew field of right quotients of the ring \((H_{\alpha},D,\rho,\sigma)\). Thus the subring \((H,D,\rho,\sigma)\) has a skew field of right quotients \(L\).

Construct, as above, a ring of right quotients \(L_1\) of the ring \((G,D,\rho,\sigma)\) with respect to the subring \((H,D,\rho,\sigma)\). It is clear that \(L\subseteq L_1\). Then every element of the ring \(L_1\) can be written in the form of a finite sum \(\sum t_{k_i}x_i y_i^{-1}\), where \(x_i,y_i\in (H,D,\rho,\sigma)\), \(k_i\in \Pi(G/H)\). Denote by \(\bar{k}_i\) the coset \(k_iH\). We shall call a formal infinite sum \(\sum t_{k_i}x_i y_i^{-1}\) an \(L\)-series if the set of elements \(\bar{k}_i\) for which the coefficients \(x_i y_i^{-1}\ne 0\) standing at \(t_{k_i}\) is well ordered in decreasing order in the sense of the prescribed ordering of the group \(G/H\). Adding \(L\)-series by the usual rules, we again obtain an \(L\)-series. In order to multiply two formal sums \(\sum t_{k_i}a_i,\ \sum t_{h_i}b_i\), where \(a_i,b_i\in L,\ k_i,h_i\in \Pi(G/H)\), one must multiply each term of the first sum by each term of the second, as defined in \(L_1\), write the resulting products in the form of a formal sum, and collect like terms. It is easy to check that the product of two \(L\)-series is defined and is an \(L\)-series, and therefore the set of \(L\)-series will be a ring. We show that each of its elements is invertible. For this it is enough to construct an inverse element for an \(L\)-series of the form \(t_1\rho_{1,1}^{-1}+\sum t_{k_i}a_i=t_1\rho_{1,1}^{-1}+u\). By A. I. Mal’cev’s lemma (2), the series \(t_1\rho_{1,1}^{-1}-u+u^2-u^3+\cdots\) makes sense and is an \(L\)-series. Then

\[ (t_1\rho_{1,1}^{-1}+u)(t_1\rho_{1,1}^{-1}-u+u^2-u^3+\cdots)=t_1\rho_{1,1}^{-1}. \]

The theorem is proved.

Let us note that if the group \(H\) is right-orderable (and \(RN^*\)-groups with torsion-free factors have this property), then, by (3), the ring \((H,D,\rho,\sigma)\) contains no zero divisors.

In conclusion I express my gratitude to Prof. A. G. Kurosh for the help given to me.

Moscow State University
named after M. V. Lomonosov

Received
2 III 1963

REFERENCES

1. M. Ikeda, Osaka Math. J., 14, 144 (1962).
2. A. I. Mal’cev, DAN, 60, 1500 (1948).
3. A. A. Bovdi, DAN, 137, 1267 (1961).
4. N. Jacobson, Lie Algebras, N. Y.—London, 1962.