On the Tzintzis radical (Q1911357)

G. Tzintzis introduced an interesting radical property \(\mathcal X\), which is the upper radical property determined by the class of all rings without admissible ideals which are not ideals, and put the question whether \(\mathcal X\) coincides with the class \(I_{\mathcal L}\) of all idempotent \(\mathcal L\)-radical rings, where \(\mathcal L\) is the Levitzki radical. In this note we will answer the question in the negative by constructing a commutative, non-simple and idempotent *-ring, that is, a prime ring whose every proper homomorphic image is \(\mathcal B\)-radical, where \(\mathcal B\) denotes the Baer lower radical.