Tumurbat, S.; Zand, H. Hereditariness, strongness and relationship between Brown-McCoy and Behrens radicals. (English) Zbl 0978.16022 Beitr. Algebra Geom. 42, No. 1, 275-280 (2001). The authors explore the properties of being hereditary and being strong among the radicals of associative rings. They also prove certain results concerning the relationship between the Brown-McCoy and Behrens radicals.If \(A\) is an associative ring then \(I\triangleleft A\), \(L\triangleleft_l A\), and \(R\triangleleft_r A\) denote that \(I\) is an ideal, \(L\) is a left ideal, and \(R\) is a right ideal of \(A\), respectively. \(A^0\) will stand for the ring on the additive group \((A,+)\) with multiplication \(xy=0\) for all \(x,y\in A\).If \(\mathcal M\) is a class of rings, let \(\overline{\mathcal M}=\{A\mid\text{every ideal of }A\) is in \({\mathcal M}\}\). A radical \(\gamma\) is principally left hereditary if \(a\in A\in\gamma\) implies \(Aa\in\gamma\). A radical \(\gamma\) is left strong if \(L\triangleleft_l A\) and \(L\in\gamma\) imply \(L\subseteq\gamma(A)\). A radical \(\gamma\) is normal if \(\gamma\) is left strong and principally left hereditary.Consider the following condition a left ideal \(L\) of a ring \(A\) may satisfy with respect to a class \(\mathcal M\) of rings: \[ L\triangleleft_l A\text{ and }Lz\in{\mathcal M} \forall z\in L\cup\{1\}.\tag{*} \] A radical \(\gamma\) is principally left strong if \(L\subseteq\gamma(A)\) whenever the left ideal \(L\) of a ring \(A\) satisfies (*) with respect to the class \(\gamma\). Principally right hereditary, right strong, and principally right strong are defined analogously. Let \(\overline\gamma\) denote the largest hereditary subclass of \(\gamma\). Let \(\gamma_l=\{A\in\gamma\mid\) every left ideal of \(A\) is in \(\gamma\}\) and \(\gamma_r=\{A\in\gamma\mid\) every right ideal of \(A\) is in \(\gamma\}\).The authors focus on two conditions that a class \(\mathcal M\) can satisfy.(H) If \(A^0\in{\mathcal M}\) then \(S\in{\mathcal M}\) for every subring \(S\subseteq A^0\).(Z) If \(A\in{\mathcal M}\) then \(A^0\in{\mathcal M}\).The lower principally left strong radical construction \({\mathcal L}_{ps}({\mathcal M})\) is similar to the lower strong radical construction \({\mathcal L}_s({\mathcal M})\). Let \(\mathcal M\) be a homomorphically closed class of rings and let \({\mathcal M}={\mathcal M}_1\), \({\mathcal M}_{\alpha+1}=\{A\mid\) every nonzero homomorphic image of \(A\) has a nonzero left ideal with (*) in \({\mathcal M}_\alpha\) or a nonzero ideal \(I\in{\mathcal M}_\alpha\}\) for ordinals \(\alpha\geq 1\) and \({\mathcal M}_\lambda=\bigcup_{\alpha<\lambda}{\mathcal M}_\alpha\) for limit ordinals \(\lambda\). The class \({\mathcal L}_{ps}({\mathcal M})=\bigcup_\alpha{\mathcal M}_\alpha\).The authors prove the following: Theorem. Let \(\gamma\neq 0\) be a principally left strong radical with (Z) and (H). Then \(\gamma_r\) is contained in \(\gamma\) as a largest nonzero hereditary and normal subradical. Furthermore, if \(\overline\gamma\) is contained in \(\gamma\) as a largest nonzero hereditary principally left strong subradical.As a result of this theorem, it can be seen that the largest left hereditary subclass \(S_l\) of a strongly prime radical \(S\) is the largest normal radical contained in \(S\).The authors prove the following statements are equivalent for a radical \(\gamma\): (i) \(\gamma\) is hereditary and normal. (ii) \(\gamma\) is left or right principally hereditary, principally left or right strong and satisfies condition (H). (iii) There exists a principally left (right, respectively) strong radical \(\delta\) such that \(\delta_r=\gamma\) (\(\delta_l=\gamma\), respectively) and satisfies conditions (Z) and (H). (iv) There exists a right (left, respectively) hereditary class \(\mathcal M\) of rings satisfying (Z) such that \(\gamma={\mathcal L}_{ps}({\mathcal M})\) (\(\gamma={\mathcal L}_{ps}'({\mathcal M})\), respectively) where \({\mathcal L}_{ps}'({\mathcal M})\) is a principally right strong radical generated by \(\mathcal M\).Finally, the authors prove the following: If \(B\) is the Behrens radical class and \(\mathcal G\) is the Brown-McCoy radical class then \({\mathcal L}_{ps}({\mathcal G})={\mathcal L}_{ps}(B)\vee{\mathcal G}\) and \({\mathcal G}_2=B_2\vee{\mathcal G}\). Reviewer: Rebecca Slover-Crittenden (Blacksburg) MSC: 16N80 General radicals and associative rings Keywords:principally left hereditary radicals; left strong radicals; lower principally left strong radicals; homomorphically closed classes of rings; normal subradicals; strongly prime radicals; Behrens radical; Brown-McCoy radical PDFBibTeX XMLCite \textit{S. Tumurbat} and \textit{H. Zand}, Beitr. Algebra Geom. 42, No. 1, 275--280 (2001; Zbl 0978.16022) Full Text: EuDML EMIS