A note on compact elements in the lattice of radicals (Q1401625)

An element \(c\) of a complete lattice \(L\) is called compact if \(c\leqq\bigvee X\) implies \(c\leqq\bigvee Y\) for some finite subset \(Y\) of \(X\) whenever \(X\) is a subset of \(L\). Compact elements of the complete lattice of all radicals of associative rings are called compact radicals. An ideal \(I\) of a ring \(A\) is called a radical ideal if \(I\) is equal to the radical \(\alpha(A)\) of \(A\), for some radical \(\alpha\). A ring \(A\) is called radical compact if the lower radical \(l_A\) determined by the class \(\{A\}\) is a compact radical. In this interesting paper the author constructs a radical compact ring \(A\) whose Dorroh extension \(A^1\) is not radical compact. This enables the author to answer two questions put by J. E. van den Berg during the conference ICOR 2000. Readers will find few minor language mistakes in this paper.