UDC 519.48

MATHEMATICS

V. P. ELIZAROV

ON THE MULTIPLICATIVE CLOSEDNESS OF A SYSTEM OF ELEMENTS OF A RING OF QUOTIENTS

(Presented by Academician V. M. Glushkov, 24 IV 1968)

Professor H. J. Weinert drew the author’s attention to the fact that the system of elements \(S_{1(S_2)}\) of the ring \(R_{(S_2)}\) need not be multiplicatively closed (see \((^1)\)). Thus, the results of \((^1)\) are valid only under the condition that the system \(S_{1(S_2)}\) is multiplicatively closed. Below we shall give a condition that is sufficient for this.*

A multiplicatively closed system \(S\) of elements of a ring \(R\), not containing zero (an m.c. system), is called a maximal m.c. system if it is not contained in any other m.c. system. For an m.c. system \(S\), by \(\{S,r\}\), where \(r \in R \setminus S\), we shall denote the least m.c. system containing \(S\) and \(r\), if such a system exists. The elements of the system \(\{S,r\}\) have the form \(s_0 r s_1 r \ldots r s_n\), where \(s_i \in S\), and some of the \(s_i\) may be absent.

Let \(S\) be an m.c. system. A two-sided ideal \(I\) of the ring \(R\) is called almost \(S\)-prime if \(I \cap S = \varnothing\) and from \(rs \in I\) or \(sr \in I\), where \(r \in R\) and \(s \in S\), it follows that \(r \in I\) \((^2)\). We shall call an m.c. system \(S\) maximal with respect to an almost \(S\)-prime ideal \(I\) if the ideal \(I\) is not almost \(\{S,r\}\)-prime for any \(r \in R \setminus S\) \((^3)\).

A maximal m.c. system \(S\) is maximal with respect to any almost \(S\)-prime ideal. The converse is not true. In the ring \(Z/(6)\), take two m.c. systems \(S=\{\overline{1},\overline{5}\}\) and \(S_1=\{\overline{1},\overline{2},\overline{4},\overline{5}\}\). The ideal 0 is almost \(S\)-prime, since there are no zero divisors in \(S\). Any m.c. system \(\{S,r\}\), where \(r \in R \setminus S\), already contains zero divisors, i.e. 0 is not an almost \(\{S,r\}\)-prime ideal, and the m.c. system \(S\) is maximal with respect to the ideal 0. At the same time \(S \subset S_1\), i.e. the m.c. system \(S\) is not maximal.

Lemma. An m.c. system \(S\) of a ring \(R\) is maximal with respect to an almost \(S\)-prime ideal \(I\) if and only if the set of all non-zero-divisors in \(\varphi(R) \cong R/I\) coincides with \(\varphi(S)\) (see \((^3)\)).

Proof. Let the system \(S\) be maximal with respect to the ideal \(I\), and let \(\varphi(r)\) be a non-zero-divisor in \(\varphi(R)\). Then from \(r_1 r \in I\) or \(r r_1 \in I\), where \(r_1 \in R\), it follows that \(r_1 \in I\). Suppose \(r \notin S\) and there exists an m.c. system \(\{S,r\}\). Since the ideal \(I\) is no longer almost \(\{S,r\}\)-prime, either \(\{S,r\}\cap I \ne \varnothing\), or from the relation \(s' r \in I\) (or \(r s' \in I\)), where \(s' \in \{S,r\}\), it does not follow that \(r \in I\).

Let \(\{S,r\}\cap I \ne \varnothing\), i.e. \(s_0 r s_1 r \ldots r s_n \in \{S,r\}\cap I\). Since \(I\) is an almost \(S\)-prime ideal, \(r s_1 r \ldots s_{n-1} r \in I\) or \(r r' \in I\), where \(r' = s_1 r \ldots r s_{n-1}\). By the preceding, \(r' \in I\), and again \(r s_2 \ldots s_{n-1} r \in I\). Proceeding similarly, we obtain \(r \in I\), which contradicts the condition.

In exactly the same way, from \(s' r \in I\) or \(r s' \in I\) it follows that \(r \in I\). Thus \(I\) is an almost \(\{S,r\}\)-prime ideal, which is impossible if \(r \in R \setminus S\). Consequently, \(\varphi(r) \in \varphi(S)\).

If, however, the m.c. system \(\{S,r\}\) does not exist, then \(s_0 r s_1 r \ldots s_{n-1} r s_n = 0\), and we can argue as above.

* Taking this occasion, the author expresses his gratitude to Professor H. J. Weinert.

Let now \(\varphi(S)\) be the set of all non-zero-divisors in \(\varphi(R)\). If \(r \in R \setminus \bar S\), then there exists an element \(r_1 \in R \setminus I\) such that \(rr_1 \in I\) or \(r_1r \in I\). If \(r \notin I\), then the second condition in the definition of an almost \(\{S,r\}\)-prime ideal is violated. If, however, \(r \in I\), then \(\{S,r\}\cap I=\varnothing\). Thus \(S\) is maximal relative to the almost \(S\)-prime ideal \(I\). The lemma is proved (see (3)).

Assertion. If \(I_1 \supset I_2\), \(S_1 \supset S_2\), and the system \(S_1\) is maximal relative to the ideal \(I_1\), then the system \(S_{1(S_2)}\) is multiplicatively closed (see (1)).

Proof. Let \(\dfrac{s_1}{s_2}, \dfrac{s'_1}{s'_2}\in S_{1(S_2)}\). Then
\[ \frac{s_1}{s_2}\frac{s'_1}{s'_2}=\frac{rs'_1}{s'_2\alpha}, \]
where \(s_1\alpha-s'_2r\in I_2\), \(r\in R\), and \(\alpha\in S_2\). Since \(I_2\subset I_1\) and \(S_2\subset S_1\), we have \(s_1\alpha\in S_1\), \(s'_2\in S_1\), and \(s-\bar s r\in I\), where \(s=s_1\alpha\), \(\bar s=s'_2\). In the ring \(\varphi_1(R)\cong R/I_1\) we have the equality \(\varphi_1(s)=\varphi(\bar s)\varphi_1(r)\), and in the ring \(R_{(S_1)}\) the element \(\varphi_1(r)=\varphi_1(\bar s)^{-1}\varphi_1(s)\) is invertible, i.e. \(\varphi_1(r)\) is a non-zero-divisor in \(\varphi_1(R)\). By the lemma, \(\varphi_1(r)\in \varphi_1(S_1)\), and \(r\in S_1\). Thus \(rs'_1\in S_1\) and
\[ \frac{s_1}{s_2}\frac{s'_1}{s'_2}\in S_{1(S_2)}. \]
The assertion is proved.

REFERENCES

\({}^{1}\) V. P. Elizarov, DAN, 135, No. 2, 252 (1960).
\({}^{2}\) V. P. Elizarov, A. I. Pilatovskaya, Siberian Mathematical Journal, 5, No. 5, 1191 (1964).
\({}^{3}\) V. P. Elizarov, A. I. Pilatovskaya, Fifth All-Union Colloquium on General Algebra (summary of communications and reports), Novosibirsk, 1963, p. 20.