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On the least multiplicative nucleus of a ring. (English) Zbl 0791.16019

Let \(R\) be a ring with identity and \(\text{Idl } R\) be the ideal lattice of \(R\). A function \(k\) on \(\text{Idl } R\) is a closure operator if \(k\) is inflationary, order-preserving, and idempotent. A multiplicative nucleus on \(\text{Idl } R\) is a closure operator \(k\) on \(\text{Idl } R\) such that, for all ideals \(I\) and \(J\) of \(R\), (1) \(k(I)\cap k(J)= k(I\cap J)= k(IJ)\) and (2) if \(k(I)=R\), then \(I=R\).
It is known that the least multiplicative nucleus of a commutative ring is the Levitzki radical. In this paper, the author shows the existence of the least multiplicative nucleus for any ring by defining, through transfinite induction, a new kind of radical ideal. In commutative rings, this radical ideal is the Levitzki radical. If \(a\in R\), let (a) be the principal ideal generated by \(a\). A ring \(R\) is called an \(m^*\)-ring if for each \(a\in R\) and each natural number \(n\), \((a)^ n\) is finitely generated. The author shows that the prime ideal theorem for distributive lattices implies that each semisimple ideal is an intersection of prime ideals for \(m^*\)-rings.

MSC:

16N80 General radicals and associative rings
06D05 Structure and representation theory of distributive lattices
16D25 Ideals in associative algebras
16U80 Generalizations of commutativity (associative rings and algebras)
16N60 Prime and semiprime associative rings
06A15 Galois correspondences, closure operators (in relation to ordered sets)
13A10 Radical theory on commutative rings (MSC2000)
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[1] Banaschewski, B., The power of the ultrafilter theorem, J. London Math. Soc., 27, 2, 193-202 (1983) · Zbl 0523.03037
[2] Banaschewski, B.; Harting, R., Lattice aspects of radical ideals and choice principles, Proc. London Math. Soc., 50, 385-404 (1985) · Zbl 0569.16003
[3] Banaschewski, B., Prime elements from prime ideals, Order, 2, 211-213 (1985) · Zbl 0576.06010
[4] Blass, A., Prime ideals yield almost maximal ideals, Fund. Math., 127 (1986) · Zbl 0609.06006
[5] Johnstone, P. T., Almost maximal ideal theorem, Fund. Math., 123, 197-209 (1984) · Zbl 0552.06004
[6] Niefield, S.; Rosenthal, K. I., Constructing locales from quantales, Math. Proc. Cambridge Philos. Soc., 104, 215-234 (1988) · Zbl 0658.06007
[7] Rav, Y., Variants of Rado’s selection lemma and their applications, Math. Nachr., 79, 145-165 (1977) · Zbl 0359.02066
[8] Sun, S.-H., On separation lemmas, J. Pure Appl. Algebra, 78, 301-310 (1992) · Zbl 0759.06007
[9] S.-H. Sun, Prime spectra of non-commutative rings, Preprint.; S.-H. Sun, Prime spectra of non-commutative rings, Preprint.
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