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Hereditariness, strongness and relationship between Brown-McCoy and Behrens radicals. (English) Zbl 0978.16022

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 \({\mathcal G}_2=B_2\vee{\mathcal G}\).

MSC:

16N80 General radicals and associative rings
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