A. V. MIKHALEV

SPECIAL STRUCTURAL SPACES OF RINGS

(Presented by Academician A. I. Mal’tsev on 6 XII 1962)

A class of rings* \(\Sigma\) is called special if:

1) every ring in \(\Sigma\) is a prime ring;
2) every nonzero ideal** of a ring in \(\Sigma\) belongs to \(\Sigma\);
3) if \(K\) is a prime ring, \(A \in \Sigma\) is its nonzero ideal, then \(K \in \Sigma\).

By \(R_\Sigma\) we shall denote the special radical determined by the special class \(\Sigma\) (see \((^{2})\)), and by \(S_\Sigma(K)\) the set of \(\Sigma\)-special ideals of the ring \(K\), i.e. such ideals \(I\) that \(K/I \in \Sigma\). The set \(S_\Sigma(K)\) with the topology in which the closure of a subset \(M \subseteq S_\Sigma(K)\) is defined as
\[ \overline M=\{I;\ I\in S_\Sigma(K),\quad I\supseteq \bigcap_{B\in M} B\}, \]
is called the \(\Sigma\)-special structural space of the ring \(K\). If \(U\) is a subset of the ring \(K\), then we define
\[ Q_\Sigma(U)=\{I;\ I\in S_\Sigma(K),\ I\supseteq U\} \]
and
\[ P_\Sigma(U)=\{I;\ I\in S_\Sigma(K),\ U\nsubseteq I\}. \]

Let \(A\) be an ideal of the ring \(K\). Then, using \((^{2})\), p. 193, and repeating the corresponding arguments of Jacobson \((^{4})\), p. 298, it is easy to verify that the mappings \(I\to I/A\) and \(J\to J\cap A\), where \(I\in Q_\Sigma(A)\) and \(J\in P_\Sigma(A)\), are homeomorphic mappings of \(Q_\Sigma(A)\) onto \(S_\Sigma(K/A)\), and of \(P_\Sigma(A)\) onto \(S_\Sigma(A)\), respectively. In particular, the mapping \(I\to I/R_\Sigma(K)\), where \(I\in S_\Sigma(K)\), is a homeomorphism of \(S_\Sigma(K)\) onto \(S_\Sigma(K/R_\Sigma(K))\).

Theorem 1. The \(\Sigma\)-special structural space \(S_\Sigma(K)\) of the ring \(K\) is bicompact if and only if for every ideal \(A\) in \(K\) such that \(K/A\) is an \(R_\Sigma\)-radical ring, there exists a finitely generated ideal \(I\in A\) such that \(A/I\) is also an \(R_\Sigma\)-radical ring.

For the proof it suffices to apply the structure theorem of Blair and Eagan \((^{6})\) and the simple observation that an ideal \(A\) of the ring \(K\) belongs to no ideal \(I\) from \(S_\Sigma(K)\) if and only if \(K/A\) is an \(R_\Sigma\)-radical ring.

Corollary 1. If every ideal \(I\) of the ring \(K\) is finitely generated, then \(S_\Sigma(K)\) is bicompact for any special class of rings \(\Sigma\) (cf. \((^{7})\), theorem 2).

Corollary 2. If \(S_\Sigma(K)\) is bicompact, then \(S_\Sigma(K)\) is homeomorphic to \(S_\Sigma(I)\), where \(I\) is some finitely generated ideal of the ring \(K\) and \(K/I\) is an \(R_\Sigma\)-radical ring (cf. \((^{7})\), theorem 3), since \(S_\Sigma(K)=P_\Sigma(I)\) and \(P_\Sigma(I)\) is homeomorphic to \(S_\Sigma(I)\).

Corollary 3. If \(K\) is a ring that is not mapped homomorphically onto nonzero \(R_\Sigma\)-radical rings, then \(S_\Sigma(K)\) is bicompact if and only if \(K\) is generated as an ideal by a finite number of elements.

Proposition 1. Let \(\Sigma\) be a special class of rings, and \(R_\Sigma\) the corresponding special radical. The condition \(R_\Sigma \leq R_2\) (see \((^{1})\)), where \(R_2\) is the Brown–McCoy radical, is necessary and sufficient

* Only associative rings are considered.
** By an “ideal,” everywhere unless otherwise stated, a two-sided ideal is meant.

for the \(\Sigma\)-special structure space \(S_\Sigma(K)\) of every ring \(K\) with \(1\) to be a nonempty bicompact space. The proof follows from Corollary 3 and the characterization of \(R_2\)-radical rings as rings that are not homomorphically mapped onto nonzero rings with \(1\).

One can verify (by analogy with \((^4)\), p. 302, Proposition 2) that there is a dependence between direct decompositions of the ring \(K\) into ideals and the connectedness of the space \(S_\Sigma(K)\) for the case when \(K\) is an \(R_\Sigma\)-semisimple ring that is not homomorphically mapped onto nonzero \(R_\Sigma\)-radical rings.

Consider the condition on the ring \(K\):

C. The structure of ideals \(L\) of the ring \(K\) has complements.

Remark 1. Let \(\Sigma\) be a special class of rings and let \(K\) be an \(R_\Sigma\)-semisimple ring satisfying condition C. Then the \(\Sigma\)-special ideals are characterized as the complements of minimal ideals of the ring \(K\) in the structure \(L\).

The proof in \((^8)\), given when \(\Sigma\) is the class of primitive rings, is also valid in the general case.

Remark 2. If a ring \(K\) is semisimple with respect to a hereditary radical \(R\) (see \((^2)\)) and satisfies condition C, then \(K\) is strongly \(R\)-semisimple, i.e., all nonzero homomorphic images of the ring \(K\) are also \(R\)-semisimple.

Remark 3. If a ring \(K\) is not homomorphically mapped onto nonzero \(R_\Sigma\)-radical rings and satisfies condition C, then every proper ideal \(A\) of the ring \(K\) is contained in some \(\Sigma\)-special ideal \(I\).

Theorem 2. If \(\Sigma\) is a special class of rings and \(K\) is a strongly \(R_\Sigma\)-semisimple ring, then the structure \(L\) of ideals of the ring \(K\) is isomorphic to the structure of open sets of the \(\Sigma\)-special structure space \(S_\Sigma(K)\) of the ring \(K\) and, consequently, is distributive.

If \(I\) is an ideal in \(K\), then \(K/I\) is \(R_\Sigma\)-semisimple and therefore
\[ I=\bigcap_{J\in Q_\Sigma(I)} J . \]

The mapping \(G: I \to Q_\Sigma(I)\) is a dual isomorphism of the structure \(L\) onto the structure of closed sets of the space \(S_\Sigma(K)\). The desired isomorphism is realized by the mapping
\[ I \to S_\Sigma(K)\setminus G(I). \]

Corollary 4. Theorem 2 holds if \(K\) is an \(R_\Sigma\)-semisimple ring satisfying condition C (cf. \((^8)\), Theorem 3). In this case the structure \(L\) of ideals of the ring \(K\) is a Boolean algebra.

Theorems 3 and 4 were proved in \((^8)\) for the case when \(\Sigma\) is the class of primitive rings. The same arguments are applicable to an arbitrary special class of rings \(\Sigma\).

Theorem 3. If a ring \(K\) satisfies condition C, then for any special class of rings \(\Sigma\) the space \(S_\Sigma(K)\) is discrete. It is compact if and only if there exists an element \(x\in K\) that belongs to no \(\Sigma\)-special ideal of the ring \(K\).

Corollary 5. A ring \(K\) is a direct sum of a finite number of simple \(\Sigma\)-special rings if and only if it is \(R_\Sigma\)-semisimple, satisfies condition C, and its \(\Sigma\)-special structure space \(S_\Sigma(K)\) is bicompact.

Theorem 4. If every proper ideal of an \(R_\Sigma\)-semisimple ring \(K\) is contained in some \(\Sigma\)-special ideal and the space \(S_\Sigma(K)\) is discrete, then \(K\) satisfies condition C.

Remark 4. In an \(R\)-semisimple ring \(K\), where \(R\) is the upper nilpotent radical (see \((^2)\)), condition Cr (the requirement that the structure of right ideals have complements) implies condition C. Therefore Theorem 3 is true with condition C replaced by condition Cr.

If \(R\) is a special radical, then it can be determined by several different special classes \(\Sigma_\alpha\), i.e., \(R_{\Sigma_\alpha}=R\), among which there exists a greatest special class \(\Sigma_0\), consisting of all

primary \(R\)-semisimple rings (see (2)). In the ring \(K\) one may consider the spaces \(S_{\Sigma_\alpha}(K)\), among which the largest will be \(S_{\Sigma_0}(K)\), and every \(S_{\Sigma_\alpha}(K)\) is everywhere a dense subset of \(S_{\Sigma_0}(K)\). Fixing a certain special class \(\Sigma_\alpha\), we can characterize the class \(\Sigma_0\) among the \(R\)-semisimple rings (cf. \((^{10})\)).

Theorem 5. An \(R\)-semisimple ring \(K\), where \(R\) is a special radical, belongs to the largest special class \(\Sigma_0\) such that \(R_{\Sigma_0}=R\) if and only if every proper closed subset of the space \(S_{\Sigma_{\alpha_0}}(K)\) is nowhere dense, i.e. its complement is everywhere dense.

Proof (cf. \((^{10})\), Theorem 11). Let \(K\in\Sigma_0\), \(\bar S_1=S_1\ne S_{\Sigma_{\alpha_0}}(K)\), \(S_2=S_{\Sigma_{\alpha_0}}(K)\setminus S_1\). If we put
\[ K_1=\bigcap_{I\in S_1} I \]
and
\[ K_2=\bigcap_{I\in S_2} I, \]
then from \(K_1\cdot K_2\subseteq K_1\cap K_2=0\) and \(K_1\ne0\) it follows that \(K_2=0\), i.e. \(\bar S_2=S_{\Sigma_{\alpha_0}}(K)\). Now let \(K\) be an \(R\)-semisimple ring in which every proper closed subset of the space \(S_{\Sigma_{\alpha_0}}(K)\) is nowhere dense, and let \(A\) be a nonzero ideal of the ring \(K\). Put
\[ S_1=\{I,\ I\in S_{\Sigma_{\alpha_0}}(K),\ I\supset A^*\} \]
and
\[ S_2=\{I;\ I\in S_{\Sigma_{\alpha_0}}(K),\ I\supset A,\ A^*\nsubseteq I\}, \]
where \(A^*\) is the annihilator of \(A\). It is obvious that \(S_1\cap S_2=\varnothing\), \(S_1\cup S_2=S_{\Sigma_{\alpha_0}}(K)\), and \(S_1\) is a closed subset. If \(S_1\ne S_{\Sigma_{\alpha_0}}(K)\), then, by assumption,
\[ \bigcap_{I\in S_2} I=0, \]
i.e. \(A=0\). Thus \(S_1=S_{\Sigma_{\alpha_0}}(K)\), and then
\[ \bigcap_{I\in S_1} I=0, \]
i.e. \(A^*=0\) for any ideal \(A\). Consequently, the ring \(K\) is primary, i.e. \(K\in\Sigma_0\).

Examples of special classes of rings are: fields, subrings of fields, rings without zero divisors, matrix rings over fields, simple rings with \(1\) (Brown–McCoy radical), primitive rings (Jacobson radical), subdirectly irreducible rings with idempotent core (Andrunakievich antisimple radical), primary rings without locally nilpotent ideals (Levitzki radical), all primary rings (Baer–McCoy radical).

The number of examples can be increased if one takes into account that the intersection and union of two special classes, as well as the complement of a special class to the class of all primary rings, are special classes of rings.

Moscow State University
named after M. V. Lomonosov

Received
1 XII 1962

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