On the least multiplicative nucleus of a ring
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.
scientific article; zbMATH DE number 89556
scientific article