Principally left hereditary and principally left strong radicals (Q2773011)

A radical \(\gamma\) is principally left hereditary if \(a\in A\in\gamma\) implies \(Aa\in\gamma\). A radical \(\gamma\) is left strong if \(L\vartriangleleft_lA\) and \(L\in\gamma\) imply \(L\subseteq\gamma(A)\), where \(L\vartriangleleft_lA\) means that \(L\) is a left ideal of a ring \(A\) and \(\gamma(A)\) denotes the \(\gamma\)-radical of \(A\). A radical \(\gamma\) is principally left strong if \(\gamma(L)=L\vartriangleleft_lA\) and \(Lz\in\gamma\) for all \(z\in L\) imply \(L\subseteq\gamma(A)\). A radical \(\gamma\) is normal if \(\gamma\) is both, left strong and principally left hereditary.NEWLINENEWLINENEWLINEGiven a ring \(A=(A,+,\cdot)\), the authors define \((A,+,\diamond_a)\) as the ring on the additive group \((A,+)\) with multiplication \(x\diamond_ay=x\cdot a\cdot y\), for \(x,y\in A\), where \(a\in A\) is a fixed element. They use \(A^\circ\) to denote the ring on the additive group \((A,+)\) with multiplication \(xy=0\).NEWLINENEWLINENEWLINEFor a class \(\gamma\) of rings the authors consider two classes, namely \(\gamma^\diamond=\{\text{rings }A:(A,+,\diamond_a)\in\gamma\), for every \(a\in A\}\) and \({\mathcal M}\gamma=\{\text{rings }A:Aa\in\gamma\), for every \(a\in A\}\).NEWLINENEWLINENEWLINEThey prove many interesting properties of these classes and use them to obtain various characterizations of normal radicals. In particular, they prove that a radical \(\gamma\) is normal if and only if \(\gamma\) is both, principally left strong and principally left hereditary. They also show that a radical \(\gamma\) is hereditary and normal if and only if \(\gamma\) is both, principally left strong and principally left hereditary, and satisfies condition (H): If \(A^\circ\in\gamma\), then \(S^\circ\in\gamma\) for every subring \(S^\circ\subseteq A^\circ\). For a radical \(\gamma\) which satisfies condition (H) the authors prove that \(\gamma\) is normal if and only if \(\gamma\subseteq\gamma^\diamond\) and \(\gamma\) is principally left strong. In this case, \(\gamma=\gamma^\diamond\) and \(\gamma\) is hereditary. Moreover, the authors construct various kinds of principally left strong radicals which are not left strong.NEWLINENEWLINENEWLINELet \(\mathcal B\) denote the Behrens radical, that is, the upper radical of all subdirectly irreducible rings with non-zero idempotents in their heart. Let \(\mathcal G\) stand for the Brown-McCoy radical, that is, the upper radical of all simple rings with unity element. The authors show that \({\mathcal G}^\diamond=\mathcal{MG}={\mathcal B}\) and deduce that \(\mathcal B\) is the largest principally left hereditary subclass of \(\mathcal G\). This is a stronger version of \textit{E. R. Puczyłowski}'s and \textit{H. Zand}'s result [proved in Quaest. Math. 19, No. 1-2, 47-58 (1996; Zbl 0853.16024)]. Finally, the authors prove that there are no left strong radicals in huge intervals in the lattice of all radicals, and neither \(\mathcal B\) nor \(\mathcal G\) is principally left strong.