Radicals induced by the total of rings
If \(A\) is a ring, the total \(\text{Tot}(A)\) is defined as the set \(\{a\in A\mid aA\) does not contain a nonzero idempotent\}. Regretfully, \(\text{Tot}(A)\) need not be an ideal because it is not closed under addition. To overcome this difficulty, the authors consider the upper radical \(\mathcal K\) of all rings \(A\) with \(\text{Tot}(A)=0\). They call \(\mathcal K\) the Kasch radical and show that \(\mathcal K\) is a supernilpotent normal radical. The authors also study the upper radical \({\mathcal K}_p\) of all prime rings \(A\) with \(\text{Tot}(A)=0\). They show that \({\mathcal K}_p\) is a special normal radical containing the Kasch radical \(\mathcal K\). They then show that \({\mathcal K}\neq{\mathcal K}_p\). In the final part of the paper, the authors position the radicals \(\mathcal K\) and \({\mathcal K}_p\) in the lattice of all radicals. The Kasch radical is bigger than the Behrens and the Jacobson radicals, and it is incomparable with the Brown-McCoy radical, the generalized nil radical, and the strongly prime radical.
scientific article; zbMATH DE number 1018696