Avoidance principle and intersection property for a class of rings. (English) Zbl 1524.13033
The present paper begins by proving the following generalization of the main result of a recent paper by C. Gottlieb [J. Commut. Algebra 12, No. 1, 87–90 (2020; Zbl 1440.13035)]: let \(R\) be an integral domain and \(A_1,A_2,\dots,A_s\) be integral domains containing \(R\) such that each \(A_i\) is a subring of a given ring \(T\), for \({1\leq i\leq s}\).
If \(A\) is any subring of \(T\) of the form \(S^{-1}R\), for some multiplicative closed subset \(S\) of \(R\), such that \(A\subseteq\bigcup_{1\leq i\leq s}A_i\), then \(A\subseteq A_i\) for some \(i\).
Then, the authors consider the following situation. Let \(R\) be a ring such that Nil(\(R\)) = Z(\(R\)) (i.e., the set of all nilpotent elements coincides with the set of all zero divisors), they say that \(R\) satisfies the avoidance principle for localizations if the following holds: let \(\{A_\alpha:\alpha\in\Lambda\}\) be a family of rings of the form \(S_\alpha^{-1}R\), where \(S_\alpha\)’s are multiplicative closed subsets of \(R\), and let \(A\) be a ring of the form \(S^{-1}R\), for some multiplicative closed subset \(S\) of \(R\), such that \(A\subseteq\bigcup_{\alpha\in\Lambda}A_{\alpha}\), then \(A\subseteq A_{\alpha}\) for some \(\alpha\in\Lambda\).
Among other results that are demonstrated in this paper, I like to point out the following two.
(1) \(R\) satisfies the avoidance principle for localizations if and only if, for any family of prime ideals \(\{\mathfrak p_\alpha:\alpha\in\Lambda\}\) of \(R\), such that \(\bigcap_{\alpha\in\Lambda}\mathfrak p_\alpha\subseteq\mathfrak p\) for some prime ideal \(\mathfrak p\) of \(R\), then \(\mathfrak p_\alpha\subseteq\mathfrak p\) for some \(\alpha\in\Lambda\).
(2) Given a subring \(V\) of \(R\), \(V\) is called compact in \(R\) if, given a family \(\{V_\alpha:\alpha\in\Lambda\}\) of subrings in \(R\) such that \(V\subseteq\bigcup_{\alpha\in\Lambda}V_\alpha\), then \(V\subseteq V_\alpha\) for some \(\alpha\in\Lambda\). Then, a subring \(V\) of \(R\) is compact in \(R\) if and only if there exists an element \(x\in V\) such that the subrings of \(R\) which contains \(x\) must contain \(V\).
Then, the authors consider the following situation. Let \(R\) be a ring such that Nil(\(R\)) = Z(\(R\)) (i.e., the set of all nilpotent elements coincides with the set of all zero divisors), they say that \(R\) satisfies the avoidance principle for localizations if the following holds: let \(\{A_\alpha:\alpha\in\Lambda\}\) be a family of rings of the form \(S_\alpha^{-1}R\), where \(S_\alpha\)’s are multiplicative closed subsets of \(R\), and let \(A\) be a ring of the form \(S^{-1}R\), for some multiplicative closed subset \(S\) of \(R\), such that \(A\subseteq\bigcup_{\alpha\in\Lambda}A_{\alpha}\), then \(A\subseteq A_{\alpha}\) for some \(\alpha\in\Lambda\).
Among other results that are demonstrated in this paper, I like to point out the following two.
(1) \(R\) satisfies the avoidance principle for localizations if and only if, for any family of prime ideals \(\{\mathfrak p_\alpha:\alpha\in\Lambda\}\) of \(R\), such that \(\bigcap_{\alpha\in\Lambda}\mathfrak p_\alpha\subseteq\mathfrak p\) for some prime ideal \(\mathfrak p\) of \(R\), then \(\mathfrak p_\alpha\subseteq\mathfrak p\) for some \(\alpha\in\Lambda\).
(2) Given a subring \(V\) of \(R\), \(V\) is called compact in \(R\) if, given a family \(\{V_\alpha:\alpha\in\Lambda\}\) of subrings in \(R\) such that \(V\subseteq\bigcup_{\alpha\in\Lambda}V_\alpha\), then \(V\subseteq V_\alpha\) for some \(\alpha\in\Lambda\). Then, a subring \(V\) of \(R\) is compact in \(R\) if and only if there exists an element \(x\in V\) such that the subrings of \(R\) which contains \(x\) must contain \(V\).
Reviewer: Marco Fontana (Roma)
MSC:
| 13A99 | General commutative ring theory |
| 13B30 | Rings of fractions and localization for commutative rings |
Citations:
Zbl 1440.13035References:
| [1] | Gottlieb, C., Finite unions of overrings of an integral domain, (to appear) in J. Commut. Algebra Available at https://projecteuclid.org/euclid.jca/1543654843 · Zbl 1440.13035 |
| [2] | Smith, W. W., A covering condition for prime ideals, Proc. Am. Math. Soc. 30 (1971), 451-452 · Zbl 0219.13004 · doi:10.1090/S0002-9939-1971-0282963-2 |
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