MATHEMATICS

V. I. ARNAUTOV

ON THE THEORY OF TOPOLOGICAL RINGS

(Presented by Academician P. S. Aleksandrov on 7 I 1964)

By a topological ring, as usual, one means an associative ring which is a Hausdorff space and in which the ring operations are continuous. By an ideal of a topological ring we shall mean a two-sided, not necessarily closed, ideal. Recall that an ideal \(I\) of a topological ring \(R\) is called topologically nilpotent if for every neighborhood of zero \(V\) of the ring \(R\) there exists a natural number \(n\) such that

\[ I^n \subseteq V. \]

Let \(R\) be an arbitrary topological ring. Denote by \(\mathfrak N(R)\) the closure of the sum of all topologically nilpotent ideals of the ring \(R\). For each ordinal number \(\alpha\) we define a closed ideal \(\mathfrak N_\alpha(R)\) as follows:

1) \(\mathfrak N_0(R)=0\).

2) Suppose that \(\mathfrak N_\alpha(R)\) has been defined for every \(\alpha<\beta\). If \(\beta\) is a limit ordinal, then put

\[ \mathfrak N_\beta(R)=\left[\sum_{\alpha<\beta}\mathfrak N_\alpha(R)\right]_R \; * . \]

If, however, \(\beta=\alpha+1\), then for \(\mathfrak N_\beta(R)\) we take the inverse image in \(R\) of the ideal \(\mathfrak N(\overline R)\) of the ring \(\overline R=R/\mathfrak N_\alpha(R)\). There exists an ordinal \(\tau\) such that

\[ \mathfrak N_\tau(R)=\mathfrak N_{\tau+1}(R). \]

We shall call

\[ L(R)=\mathfrak N_\tau(R) \]

the topological Baer radical.

From the construction of \(L(R)\) it is clear that \(R/L(R)\) contains no topologically nilpotent ideals.

We shall call a ring \(R\) \(L\)-radical if \(L(R)=R\). Finally, we shall call a ring \(R\) \(L\)-semisimple if \(L(R)=0\), i.e., if \(R\) contains no topologically nilpotent ideals.

We have proved that:

1) In an arbitrary topological ring \(R\), \(L(R)\) is the intersection of all such closed ideals \(N\) of the ring \(R\) for which the quotient rings \(R/N\) contain no nonzero topologically nilpotent ideals.

2) For any ideal \(I\) of a topological ring \(R\) one has

\[ L(I)\subseteq I\cap L(R). \]

It is known that, in the discrete case, for any ideal \(I\) of a ring \(R\),

\[ L(I)=I\cap L(R). \]

This equality also holds for a topological ring \(R\) possessing a complete system of group neighborhoods of zero (see \((^6)\)). However, as follows from Theorems 2 and 3 of the present work, there exists a topological ring \(\widehat R\) and an ideal \(R'\) in \(\widehat R\) such that

\[ L(R')=0\subset R'\cap L(\widehat R)=R'. \]

In \((^6)\) it is proved that if \(R\) is an everywhere dense subring of a topological ring \(\widehat R\), then

\[ L(R)\subseteq R\cap L(\widehat R). \]

From Theorems 4 and 5 of the present work it follows that there exist rings \(\widehat R\) and \(R'+R''\) such that \(R'+R''\) is an everywhere dense subring of the ring \(\widehat R\), and

\[ L(R'+R'')=0\subset R'+R''\cap L(\widehat R)=R'. \]

\[ \text{* If } A \text{ is some subset of the topological space } R,\text{ then by } [A]_R \text{ we shall denote the closure of } A \text{ in } R. \]

It is known that for a discrete ring \(L(R)\) is a nil-ideal, and in a commutative discrete ring \(L(R)\) coincides with the set of all nilpotent elements of the ring \(R\).

In [6] it is shown that in a commutative bounded\(*\) ring \(L(R)\) coincides with the set of all topologically nilpotent elements. One can show that if a topological ring \(R\) has a full system of ideal neighborhoods of zero, then all elements of \(L(R)\) are topologically nilpotent.

However, not in every topological ring \(R\) are the elements of \(L(R)\) topologically nilpotent, even if \(R\) is a commutative ring. Thus, from Theorems 6 and 7 of the present paper it follows that there exists a commutative \(L\)-radical ring with identity.

Example 1. Let \(R'\) be the freely generated ring with a countable number of generators \(x_1, x_2,\ldots\). The elements of the ring \(R'\) are all possible finite sums of words \(x_{i_1}x_{i_2}\cdots x_{i_n}\). It is easy to verify that there are countably many such words. Consequently, all words can be numbered in a sequence \(y_1,y_2,\ldots\). Every element \(a\in R\) can be represented uniquely in the form \(\sum_{i=1}^{\infty}\alpha_i y_i\), where the \(\alpha_i\) are integers and only a finite number of \(\alpha_i\ne 0\).

We shall say that the element \(a'=\sum_{i=1}^{\infty}\alpha'_i y_i\) is smaller than the element \(a=\sum_{i=1}^{\infty}\alpha_i y_i\), and shall write \(a'<a\), if \(\alpha_i=0\) implies \(\alpha'_i=0\) and there exists an \(i_0\) such that \(\alpha'_{i_0}=0\), but \(\alpha_{i_0}\ne 0\). To each element \(a\in R'\) we associate the number \(s(a)\), equal to the degree of \(a\) regarded as a polynomial in the \(x_i\), and to each element \(0\ne a\in R'\) we associate the number
\[ m(a)=\min\{s(a')\mid 0\ne a'\leqslant a\}. \]
Clearly, \(m(a)\) is the least degree of the words \(y_i\) for which \(\alpha_i\ne 0\). We shall put \(m(0)=0\).

To any sequence of integers \(\alpha_1,\alpha_2,\ldots\), in which only a finite number of \(\alpha_i\ne 0\), and to any natural number \(n\), we associate the set \(A^{(n)}_{\alpha_1,\alpha_2,\ldots}\) of all elements \(a\in R\) which can be represented in the form
\[ \sum_{j=1}^{k} a_j b_j+\sum_{i=1}^{\infty}\alpha_i y_i, \]
such that the following conditions are satisfied: a) for any \(a'_j\leqslant a_j\), \(b'_j\leqslant b_j\), \(j=1,2,\ldots,k\),
\[ \max\{s(a'_1b'_1),s(a'_2b'_2),\ldots,s(a'_kb'_k)\}\geqslant pn, \]
where \(p\) is the number of the nonzero \(a'_j b'_j\); b)
\[ m\left(\sum_{i=1}^{\infty}\alpha_i y_i\right)\geqslant n\min(1,\sum|\alpha_i|). \]

Let \(Q\) be the discrete direct sum of fields of real numbers. Then \(Q\) is a regular ring\(**\). Denote by \(R''\) the ring of all infinite matrices over \(Q\) in which only finitely many elements are different from zero. Since the ring of all matrices of order \(n\) over a regular ring is a regular ring, and every element of \(R''\) may be regarded as a matrix of order \(n\), it follows that \(R''\) is a regular ring. From the fact that every ideal of a regular ring is a regular ideal, it easily follows that \(R''\) is \(L\)-semisimple in any topology.

For an arbitrary natural number \(n\) and an arbitrary sequence of integers \(\alpha_1,\alpha_2,\ldots\), in which only finitely many \(\alpha_i\ne 0\), define the set \(B^{(n)}_{\alpha_1,\alpha_2,\ldots}\) of all matrices \(\|\mathfrak A_{ij}\|\) for which
\[ \sum_{i\geq n}\mathfrak A_{2i+1,2i}=\alpha_1,\alpha_2,\ldots, \]
and the remaining \(\mathfrak A_{ij}=0\).

\(*\) A ring \(R\) is called bounded if for every neighborhood of zero \(V\) there exists a neighborhood of zero \(U\) such that \(U\cdot R=\{ab\mid a\in U,\ b\in R\}\subseteq V\) and \(R\cdot U\subseteq V\).

\(**\) A ring \(R\) is called regular if the equation \(x\xi x-x=0\) is solvable in \(R\) for arbitrary \(x\).

Consider the ring with zero multiplication \(R'''\) over the discrete direct sum of groups of integers. The elements of the ring \(R'''\) are all possible sequences of integers \(\mathfrak A=\alpha_1,\alpha_2,\ldots\), in which only a finite number of \(\alpha_i\ne 0\).

Let \(\hat R\) be the ring equal to the direct sum of the rings \(R'\), \(R''\), \(R'''\). The elements of the ring \(\hat R\) are all possible triples of elements \((a,\|\mathfrak A_{ij}\|,\mathfrak A)\), where \(a\in R'\), \(\|\mathfrak A_{ij}\|\in R''\), \(\mathfrak A\in R'''\), and the operations are performed componentwise.

Denote by \(V_n\) the set of all elements \((a,\|\mathfrak A_{ij}\|,\mathfrak A)\) for which \(a\in A_{\alpha_1,\alpha_2,\ldots}^{(n)}\), \(\|\mathfrak A_{ij}\|\in B_{\beta_1,\beta_2,\ldots}^{(n)}\), and \(\alpha_1+\beta_1,\alpha_2+\beta_2,\ldots=\mathfrak A\).

Theorem 1. The collection \(\{V_n\}\) may be taken as a full system of neighborhoods of zero, so as to make \(\hat R\) into a topological ring.

Theorem 2. In the topological ring \(\hat R\), \(L(\hat R)=R'+R'''\).

Proof. It can be shown that \(R'+R'''\) is a closed ideal in \(\hat R\). Since \(\hat R/R'+R'''\) is algebraically isomorphic to \(R''\), it follows that \(\hat R/R'+R'''\) is \(L\)-semisimple. Hence \(L(\hat R)\subseteq R'+R'''\). Conversely, it is obvious that \(R'''\subseteq L(\hat R)\). It can be shown that for every \(n\) one has
\[ (R')^n\subseteq R'''+V_n\subseteq L(\hat R)+V_n. \]
Since \(\hat R/L(\hat R)\) contains no nonzero topologically nilpotent ideals, \(R'+R'''=L(\hat R)\).

Theorem 3. The ideal \(R'\), considered as a subring of the topological ring \(\hat R\), is an \(L\)-semisimple ring.

Proof. From the construction of \(V_n\) it follows easily that
\[ V_n\cap R'=A_{0,0,\ldots}^{(n)}. \]
Consequently, the collection \(\{A_{0,0,\ldots}^{(n)}\}\) is a full system of neighborhoods of zero in \(R'\). For each element \(a\in R'\) and each natural number \(r\) we construct an element \(e_r\in (a)^r_{R'}\) such that \(e_r\notin A_{0,0,\ldots}^{(1)}\).

Let \(n\) be a number such that \(a\) can be obtained from the generators \(x_1,x_2,\ldots,x_n\). For the number \(r\) the number of words of degree not exceeding \(rs(a)\) obtained with the help of the generators \(x_1,x_2,\ldots,x_n\) is finite. Let these words be
\[ y_{i_1},y_{i_2},\ldots,y_{i_l}. \]
It is easy to verify that
\[ a^r=\sum_{j=1}^{l}\alpha_{i_j}y_{i_j}. \]
The ideal \((a)^r_{R'}\) contains the element
\[ e_r=\sum_{i=1}^{t}x_i a^r x_i,\quad \text{where } t=(2rs(a)+3)(l+1). \]
It can be shown that \(e_r\) cannot be represented in the form
\[ \sum_{i=1}^{p}a_i b_i, \]
so that condition a) for \(n=1\) is satisfied. From the arbitrariness of \(a\) and \(r\) it follows that \(R'\) contains no topologically nilpotent ideals.

Theorem 4. \(R'+R''\) is an everywhere dense ideal of the topological ring \(\hat R\).

Theorem 5. \(R'+R''\) is an \(L\)-semisimple ring.

Proof. According to (2) and Theorem 2,
\[ L(R'+R'')\subseteq (R'+R'')\cap L(\hat R)=(R'+R'')\cap(R'+R''')=R'. \]
Now let \(I\) be a topologically nilpotent ideal in \(R'+R''\). Then
\[ I\subseteq L(R'+R'')=R'. \]
Therefore \(I\) will be a topologically nilpotent ideal in \(R'\). But this contradicts Theorem 3. Consequently, \(R'+R''\) is an \(L\)-semisimple ring.

Example 2. Let \(P\) be the ring of all polynomials in a countable number of mutually commuting variables \(x_1,x_2,\ldots\) with integer coefficients. Then \(P\) is a commutative ring with identity \(1\). To each element \(a\in P\) assign the nonnegative number
\[ l(a)=\max\{i\mid a\text{ depends on }x_i\}. \]
If \(a\) does not depend on any \(x_i\), then we put \(l(a)=0\). For each natural number \(n\) define the set
\[ P_n=\{b\in P\mid l(b)<n\}. \]
It is obvious that the \(P_n\) are subrings containing the subring of integers of the ring \(P\).

Put
\[ V'_n=\sum_k\sum_{i=1}^k P_{n,k}(1-x_{n,k})^i, \]
where the sum is understood as the sum of the sets \(P_{n,k}(1-x_{n,k})^i\) in the ring \(P\). For each number \(n\) define the ideal
\[ V''_n=\sum_i Px_i^{\,n\cdot i}. \]
Denote \(V'_n+V''_n\) by \(V_n\).

Theorem 6. The collection \(\{V_n\}\) may be taken as a complete system of neighborhoods of zero in order to make \(P\) into a topological ring.

Theorem 7. The topological ring \(P\) is \(L\)-radical.

Proof. Since for any neighborhood of zero \(V_n\) and any element \(x_i\in P\) we have
\[ (x_i)_P^{\,n\cdot i}=Px_i^{\,n\cdot i}\subseteq V_n, \]
all \(x_i\in L(P)\). We now show that \(1\in L(P)\). Let \(V_n\) be an arbitrary neighborhood of zero. One can verify that \(1+V_n\ni x_n\). From the closedness of \(L(P)\) it follows that \(1\in L(P)\). Since \(L(P)\) is an ideal and contains the identity of the ring \(P\), we have \(L(P)=P\).

The author takes this opportunity to express his gratitude to V. A. Andrunakievich, under whose supervision this work was carried out.

Institute of Physics and Mathematics
Academy of Sciences of the MSSR

Received
6 II 1964

REFERENCES

\(^1\) N. Bourbaki, General Topology, Moscow, 1958.
\(^2\) B. Brown, N. McCoy, Proc. Am. Math. Soc., 1, 165 (1950).
\(^3\) R. Baer, Am. J. Math., 65, 87 (1943).
\(^4\) N. Jacobson, The Structure of Rings, IL, 1961.
\(^5\) J. Neumann, Proc. Nat. Acad. Sci. USA, 22, 707 (1936).