On the radicals of structural matrix rings
Let \(I=\{1,\dots,n\}\), and let \(\rho\) be a transitive relation on \(I\times I\). Let \(R\) be a ring. The subring \(M(\rho,R)\) of the complete matrix ring \(M_n(R)\) which consists of all matrices \(A=(a_{ij})\) such that \(a_{ij}=0\) whenever \((i,j)\notin\rho\) is called a structural matrix ring. In the current paper, the study of the radical theory of structural matrix rings is continued. Let \(\gamma\) be a (Kurosh-Amitsur) radical. Then \(\gamma^+:=\{R\mid R^0\in\gamma\}\) is an A-radical (i.e. a radical which depends only on the additive group of \(R\)), where \(R^0\) is the ring with zero multiplication built on the additive group of \(R\). Let \(\rho_s=\{(i,j)\in\rho\mid(j,i)\in\rho\}\) and \(\rho_a=\{(i,j)\in\rho\mid(j,i)\notin\rho\}\). The following result is proved: Theorem 1. \(\gamma(M(\rho,R))=M(\rho_s,\gamma(R))+M(\rho_a,\gamma(R))\) for every structural matrix ring \(M(\rho,R)\) if and only if \(\gamma\) has the matrix property and \(\gamma\subseteq\gamma^+\). This provides a converse to a result of \textit{A. D. Sands} [Quaest. Math. 13, No. 1, 77-81 (1990; Zbl 0705.16013)]. Various other results are proved, for example: Proposition 4. \(\gamma(M(\rho,R))\subseteq M(\rho,\gamma(R))\) for all structural matrix rings \(M(\rho,R)\) if and only if \(\gamma^+\subseteq\gamma\) and \(\gamma(M_k(R))\subseteq M_k(\gamma(R))\) for all full matrix rings \(M_k(R)\). Proposition 6. \(\gamma(M(\rho,R))=M(\rho,\gamma(R))\) for all \(M(\rho,R)\) if and only if \(\gamma\) is an A-radical. Proposition 7. Let \(0\neq\gamma\) be a hereditary subidempotent radical with the matrix property. Then \(\gamma(M(\rho,R))=M(\rho,\gamma(R))\) for all \(R\) if and only if \(\rho=\rho_s\). Let \(\rho^*=\{(i,j)\in\rho\mid(k,i)\in\rho\) implies \((i,k)\in\rho\) and \((j,k)\in\rho\) implies \((k,j)\in\rho\}\). Theorem 11. Let \(\gamma\) be a hereditary subidempotent radical with the matrix property. Then \(\gamma(M(\rho,R))=M((\rho^*)_s,\gamma(R))\).
scientific article; zbMATH DE number 1051734
scientific article