Compact elements in the lattice of radicals. (Q2387175)

The author investigates the radicals in the sense of Kurosh-Amitsur [see \textit{R. L. Snider}, Pac. J. Math. 40, 207-220 (1972; Zbl 0235.16006) or \textit{N. J. Divinsky}, Rings and radicals (1965; Zbl 0138.26303)]. Since the collection of all radicals is partially ordered by inclusion, the problem of this paper is to describe the compact elements in the lattice of all radicals. The radical \(R\) is said to be compact if, given any collection of radicals \(X\) such that \(R\geq\bigvee X\), we have \(R\leq\bigvee X'\) for some finite subcollection \(X'\) of \(X\). A ring \(A\) is said to be radical compact if the lower radical \(L\{A\}\) on the singleton \(\{A\}\) is compact. The main result asserts that every radical compact ring satisfies the finiteness condition with respect to hereditary radical ideals. Moreover, if a ring \(A\) satisfies the ACC on the radical ideals, then \(A\) is radical compact. Finally, the author proves that the class of all radical compact rings is closed under finite direct sums, but it is not closed under homomorphic images.