UDC 519.48

MATHEMATICS

L. A. BOKUT’

ON THE EMBEDDING OF RINGS IN FIELDS

(Presented by Academician A. I. Mal’tsev on 12 IX 1966)

I. In this note a proof is given of the following result:

Theorem. There exists an (associative) ring without zero divisors, not embeddable in a field, whose multiplicative semigroup of nonzero elements is embeddable in a group.

This is a solution of a problem posed by A. I. Mal’tsev.

II. Following (2), let us call a ring \(K\) an \(SN\)-ring if \(K\) is an algebra with 1 over a field \(F\), given in some system of generators \(\Xi\) by relations of the form

\[ w_i h_i = u_i f_i, \tag{1} \]

where \(i \in I,\ w_i, h_i, u_i, f_i \in \Xi\), satisfying the conditions:

S1) \(\{w_i,u_i\}\cap\{h_j,f_j\}=\varnothing\).

S2) The words \(w_i h_i,\ u_i f_i,\ i\in I\), are pairwise distinct.

N) If in \(K\) there is an equality \(AB=CD\), where \(A,B,C,D\) are nonzero linear combinations of generators from \(\Xi\), then either \(A=CX,\ D=XB,\ X\in K\), or there is an equality \(xy=zt\) among (1) such that the relations hold:

\[ A=aX(x+\alpha z),\quad B=(y+\beta t)Y,\quad C=X(z+\beta x),\quad D=(t+\alpha y)Ya, \tag{2} \]

where \(0\ne X,Y,a\in F,\ \alpha,\beta\in F\).

In (1) it is proved that, as a basis of the algebra \(K\), one may take the element 1 and all words in the generators not containing the left-hand sides of the relations (1) (assuming that in (1) the left- and right-hand sides are fixed). In (2) it is proved that \(SN\)-rings have no zero divisors.

By \(\overline{SN}\) we shall denote the totality of completions \(\overline K\) of rings \(K\in SN\) with respect to the natural norm. The ring \(\overline K\) is the ring of formal infinite series of the form

\[ A=\sum_{i=0}^{\infty} A_i, \]

where \(A_i\) are homogeneous elements of degree \(i\).

III. Let \(\overline K\) be an arbitrary \(\overline{SN}\)-ring. Introduce the following sets of elements of the ring \(\overline K\): \(\Gamma\) is the set of all invertible elements of the ring \(\overline K\); \(P\) is the set of elements of the form

\[ p(w,u,A)=w+uA,\quad p(u,w,C)=u+wC, \tag{3} \]

where \(wh=uf\) is an arbitrary one of the relations (1); \(A\) is an arbitrary element of \(\overline K\) such that all basic words in \(A\) do not begin with \(f\) and do not end with \(w\); all basic words in \(C\) do not begin with \(h\) and do not end with \(w\); \(Q\) is the set of elements of the form

\[ q(f,h,A)=f+Ah,\quad q(h,f,C)=h+Cf, \tag{4} \]

where \(wh=uf\) is an arbitrary one of the relations (1), and \(A\) and \(C\) are the same as above; \(R\) is the set of all irreducible elements (one from each class of associated elements), each of which is not associated with any element from \(P\cup Q\).

Let us single out in the sets \(P,Q\) the subsets \(P\subseteq P_1,\ P^*,\ Q\supseteq Q_1,\ Q^*\). Namely, into \(P_1\) and \(Q_1\) we place elements of the form (3) and (4) for which \(A\) and \(C\) are not invertible; into \(P^*, Q^*\) we place elements of the form \(P^*=\{p(w,u,A^*)\},\ Q^*=\)

\[ = \{q(f,h,A^*)\}, \]
where \(A^*\) are invertible elements. For a fixed relation

\[ wh=uf \tag{5} \]

we shall regard the sets of all noninvertible elements \(A\) and \(C\) occurring in (3) and (4), as well as the set of all invertible elements \(A^*\) from (3) and (4), as completely ordered:
\[ A_1,A_2,\ldots;\ C_1,C_2,\ldots;\ A_1^*,A_2^*,\ldots . \]
After this we introduce new notation for the elements of the sets \(P_1,Q_1,P^*,Q^*\), whose meaning will be clear from the examples:
\[ p(w,u,A_i)=p_i(w,u)=p_{iw},\qquad q(f,h,A_i^*)=q_i^*(f,h)=q_{if}^* . \]

We define the group \(\Gamma\) by a system of generators consisting of all elements of this group and by some system of relations \(X_i=1\). Denote by \(\overline{K}^{*}\) the multiplicative semigroup of nonzero elements of the ring \(\overline{K}\).

Theorem 1 \((^3)\). The semigroup \(\overline{K}^{*}\) is given in the system of generators

\[ P_1\cup Q_1\cup P^*\cup Q^*\cup R\cup\Gamma \tag{6} \]

by the following system of defining relations, constructed for all relations (5) among (1) and for all indices \(k,i,n,m\):

\[ \begin{aligned} &1.\quad p_k(w,u)X_{ki}q_i(h,f)=p_i(u,w)Y_{ik}q_k(f,h). \end{aligned} \tag{7} \]

\[ \begin{aligned} &2.\quad p_k^*(w,u)X_{ni}^*q_i(h,f)=p_i(u,w)Y_{in}^*q_n^*(f,h). \end{aligned} \tag{8} \]

\[ \begin{aligned} &3.\quad p_k(w,u)\overline{X}_{kn}q_n^*(f,h)=p_n^*(w,u)\overline{Y}_{nk}q_k(f,h). \end{aligned} \tag{9} \]

\[ \begin{aligned} &4.\quad p_n^*(w,u)X_{nm}^{**}q_m^*(f,h)=p_m^*(w,u)Y_{mn}^{**}q_n^*(f,h), \end{aligned} \tag{10} \]

where \(A_n^*-A_m^*\) is an invertible element,
\[ X_{nm}^{**}=Y_{mn}^{**}=(A_n^*-A_m^*). \]

\[ \begin{aligned} &5.\quad \alpha(a+L\alpha)=(a+\alpha L)\alpha, \end{aligned} \tag{11} \]
where \(0\ne a\in F,\ \alpha\in P_1\cup P^*\cup Q_1\cup Q^*\cup R\).

\[ \begin{aligned} &6.\quad X_i=1. \end{aligned} \tag{12} \]

Here we have used the notation:
\[ X_{ki}=(1-C_iA_k)^{-1},\qquad Y_{ik}=(1-A_kC_i)^{-1},\qquad X_{ni}^*=(1-C_iA_n^*)^{-1},\qquad Y_{in}^*=(1-A_n^*C_i)^{-1}, \]
\[ \overline{X}_{nk}=\overline{Y}_{kn}=(A_n^*-A_k)^{-1}. \]

IV. We now recall the definition of the rings \(F(Q_n)\) from \((^1)\). The ring \(F(Q_n)\), \(n\ge 2\), is an algebra with \(1\) over the field \(F\), given in the system of generators
\[ \{a_i,b_i,c_i,s_i,t_i,\ 1\le i\le n;\ v_0,v_1\} \]
by the defining relations
\[ a_is_i=c_iv_0,\qquad b_is_{i+1}=c_iv_1,\qquad b_it_{i+1}=a_it_i, \]
where \(1\le i\le n,\ s_{n+1}=s_1,\ t_{n+1}=t_1\). In \((^2)\) it was proved that the rings \(F(Q_n)\) are \(SN\)-rings. If \(F\) is a field of characteristic \(p\ne0,\ n=p^k\), then, as shown in \((^1)\), the rings \(F(Q_n)\) are not embeddable in fields. On the other hand, it will follow from what follows that the semigroups \(GF(2)(Q_n)^*\) are embeddable in groups. Therefore the ring \(GF(2)(Q_4)\) is the simplest example of a ring giving an answer to A. I. Mal’cev’s problem.

V. Let again \(\overline{K}\) be an arbitrary \(\overline{SN}\)-ring, and let \(G\) be the group of fractions of the semigroup \(\overline{K}^{*}\), i.e. the group given by the generators (6) and the relations (7)—(12). Fix some relation (5). In the group \(G\) the equalities

\[ \begin{gathered} X_{n1}^*q_{1n}Q_{1nf}^*q_{1h}^{-1}X_{ni}^{*-1}P_{1nw}^\prime \overline{Y}_{nk}q_{kf}Q_{2nf}^*q_{sf}^{-1}\overline{Y}_{ns}^{*-1}P_{2nw}^\prime,\\ P_{3nw}^\prime\overline{Y}_{nk}q_{kf}Q_{3nf}^*q_{sf}^{-1}\overline{Y}_{ns}^{*-1}P_{4nw}^\prime X_{n1}^*q_{1h}Q_{4ng}^*q_{ih}^{-1}X_{ni}^{*-1}, \end{gathered} \tag{13} \]

hold, if the following relations hold:

\[ \begin{aligned} &1.\quad P_{1nw}^*=P_{iu}P_{kw}.\\ &2.\quad Y_{ln}^*Q_{1nf}^*Y_{in}^{*-1}P_{iu}^\prime = P_{lu}^\prime Y_{lk}Q_{kf}Y_{ik}^{-1}.\\ &3.\quad P_{kw}^\prime\overline{X}_{kn}Q_{2nt}^*\overline{X}_{sn}^{*-1} = X_{ki}Q_{in}X_{si}^{-1}P_{sw}^\prime.\\ &4.\quad Q_{kf}^\prime Q_{in}^\prime=Q_{ln}^\prime Q_{sf}^\prime. \end{aligned} \tag{14} \]

1. \(P_{1u}=P^{*}_{3nw}P_{2kw}\).

2. \(P_{1iu}=P^{*}_{sw}P^{*}_{2nw}\).

3. \(P'_{1kw}X_{kl}Q_{ln}X^{-1}_{sl}=\overline{X}^{*}_{kn}Q^{*}_{3nf}\overline{X}^{*-1}_{sn}P'_{1sw}\).

4. \(Y_{ls}Q_{sf}Y^{-1}_{is}P'_{1iu}=P'_{1lu}Y^{*}_{ln}Q^{*}_{4nf}Y^{*-1}_{in}\).

5. \(P_{1sw}P_{1lu}=P^{*}_{4nw}\).

The notation introduced here will be clear from the following examples: \(Q^{*}_{1nf}=1+q^{*}_{nf}L_{1}\), \(Q^{*'}_{1nf}=1+L_{1}q^{*}_{nf}\), \(Q^{*}_{2nf}=1+q^{*}_{nf}L_{2}\), where \(L_{1},L_{2}\in \overline{K}\), \(P_{iu}=1+p_{iu}L\), \(P'_{iu}=1+Lp_{iu}\), \(L\in \overline{K}\).

In \(G\) the equalities

\[ \overline{X}^{*-1}_{sn}p^{-1}_{sw}\overline{P}^{*}_{1nw}p_{kw}\overline{X}^{*}_{kn}Q^{*}_{1nf}Y^{*}_{in}p^{-1}_{iu}P^{*}_{2nw}p_{lu}Y^{*}_{ln}Q_{2nf}= \]

\[ =Q^{*}_{3nf}Y^{*-1}_{in}p^{-1}_{iu}P^{*}_{3nw}p_{lu}Y^{*}_{ln}Q^{*}_{4nf}\overline{X}^{*-1}_{sn}p^{-1}_{sw}P^{*}_{4nw}p_{kw}\overline{X}^{*}_{kn}, \tag{15} \]

also hold, if relations analogous to those given in (14) hold.

VI. Let us give one more collection of equalities of the group \(G\). Suppose that among the defining relations of the ring \(K\) under consideration there is a triple of equalities of the form:

\[ wh=uf,\qquad w_{1}h_{1}=uf_{1},\qquad wh_{2}=w_{1}f_{2}. \tag{16} \]

Fix the equalities (16) and introduce the notation:

\[ \begin{array}{lll} \overline{p}_{nw}=w+uf_{1}L_{n}, & \overline{q}_{nf}=f+f_{1}L_{n}h, & \overline{q}_{nf_{2}}=f_{2}+h_{1}L_{n}h_{2},\\ \overline{p}_{iu}=u+wh_{2}M_{i}, & \overline{q}_{ih}=h+h_{2}M_{i}f, & \overline{q}_{ih_{1}}=h_{1}+f_{2}M_{i}f_{1},\\ \overline{p}_{kw_{1}}=w_{1}+ufS_{k}, & \overline{q}_{kf_{1}}=f_{1}+fS_{k}h, & \overline{q}_{kh_{2}}=h_{2}+h_{1}S_{k}f_{2}, \end{array} \tag{17} \]

where \(L_{n},M_{i},S_{k}\) are arbitrary nonzero elements of the ring \(\overline{K}\), not ending in \(w,u,w_{1}\), respectively. (We regard the sets \(\{L_{n}\}\), \(\{M_{i}\}\), \(\{S_{k}\}\) as completely ordered.)

In the group \(G\) the equalities

\[ P'_{1nw}X'_{nj}Q'_{\overline{nf}}{}^{-1}\overline{q}^{-1}_{in}X'^{-1}_{ni}P'_{nw}X'''_{nl}q_{lh_{2}}Q'_{\overline{nf_{2}}}{}^{-1}\overline{q}^{-1}_{kh_{2}}X'''^{-1}_{nk}= \]

\[ =X'''_{nl}q_{lh_{2}}Q'_{\overline{nf_{2}}}{}^{-1}\overline{q}^{-1}_{kh_{2}}X'''^{-1}_{nk}P'_{3nw}X'_{ni}\overline{q}_{jh}Q'_{\overline{nf}}{}^{-1}\overline{q}^{-1}_{ih}X'^{-1}_{ni}P'_{2hw}, \tag{18} \]

hold, if relations analogous to those given in (14) are satisfied.

The meaning of the notation introduced will be clear from examples:

\[ P_{1nw}=1+\overline{p}_{nw}A_{1},\quad P'_{1nw}=1+A_{1}\overline{p}_{nw},\quad A_{1}\in\overline{K},\quad X'_{ni}=(1-h_{2}M_{i}fL_{n})^{-1}, \]

\[ Y'_{in}=(1-f_{1}L_{n}h_{2}M_{i})^{-1},\quad X''_{ki}=(1-f_{2}M_{i}fS_{k})^{-1},\quad X'''_{nk}=(1-hS_{k}h_{1}L_{n})^{-1}. \]

VII. Let now \(F=GF(2)\). Fix some semigroup \(\overline{F(Q_{n})}^{\,*}\), \(n\ge 2\), and let \(G\) be its group of quotients. We shall show how, on the basis of what has preceded, it is proved that the semigroup \(\overline{F(Q_{n})}^{\,*}\) is a subsemigroup of the group \(G\). First of all, include among the defining relations of the group \(G\) the relations (13), (15), and (18). Next define a finite sequence of groups \(G_{i}\), \(1\le i\le 8\). All these groups are obtained according to the rule: among the generators of the group \(G\) some subset is singled out and all defining relations from the elements of this subset are considered. Therefore, in order to define the groups \(G_{i}\), it is enough to specify the sets of generators \(\Sigma_{i}\) of each of these groups. We have: \(\Sigma_{1}=\Xi\cup\Gamma\); \(\Sigma_{2}\) is obtained from \(\Sigma_{1}\) by adjoining all elements of the form \(p_{i}(u,w)\), \(q_{i}(h,f)\) such that they do not enter the sets (18), constructed for all relations (17) obtained from the relations of the ring \(F(Q_{n})\) by fixing the number \(i\), \(1\le i\le n\); \(\Sigma_{3}\) is obtained by adjoining to \(\Sigma_{2}\) elements of the form \(\overline{p}_{n}(u,w)\), \(\overline{q}_{n}(h,f)\), \(\overline{q}_{n}(h_{1},f_{1})\), constructed for all relations (17);

\(\Sigma_4\) is obtained from \(\Sigma_3\) by adding all elements \(p_n(w,u)\), \(q_n(f,h)\) that do not belong to the set (18); \(\Sigma_5\) consists of the elements of \(\Sigma_4\) and elements of the form \(\bar p_n(w_1,u)\), \(\bar q_n(f_1,h_1)\), \(\bar q_n(h_2,f_2)\), constructed for all triples (17); \(\Sigma_6\) is obtained by adding to \(\Sigma_5\) the elements \(\bar p_n(w,u)\), \(\bar q_n(f,h)\), \(\bar q_n(f_2,h_2)\), constructed for all triples (17); finally,

\[ \Sigma_7=\Sigma_6\cup P^*\cup Q^*,\quad \Sigma_8=\Sigma_7\cup R. \]

In the groups \(G_i\), \(1\le i\le 8\), a canonical form of elements is constructed successively, analogously to how this is done in the works \((^1)\) and, especially, \((^4)\). As a result it turns out that every word of the group \(G\) is equal to one and only one word having canonical form. Moreover, if \(W\) is a positive word of the group \(G\) (a word in the alphabet (6)), then to obtain its canonical form it is enough to carry out on \(W\) a certain number of “semigroup” transformations (transformations corresponding to the equalities (7)—(12)). This proves the required result.

In conclusion I consider it my pleasant duty to express sincere thanks to Acad. A. I. Mal’tsev, A. I. Shirshov, and A. D. Taimanov for the help and support given to the author, and also to Yu. L. Ershov for reading the manuscript of the work.

CITED LITERATURE

\(^1\) L. A. Bokut’, Algebra and Logic, Seminar, 3, 5—6, 5 (1964).
\(^2\) L. A. Bokut’, ibid., 4, 4, 25 (1965).
\(^3\) L. A. Bokut’, ibid., 4, 5, 17 (1966).
\(^4\) L. A. Bokut’, ibid., 5, 5, 3 (1966).