Minimal and maximal ideals in rings with involution
Let \(R\) be a ring with involution *. If \(I\) is a *-minimal ideal of \(R\) such that \(I^2\neq 0\), then \(I^2=I\) and \(I\) is a *-simple ring. The socle of \(R\) is a direct sum of the sum of all *-minimal ideals and of the direct sum of minimal ideals \(K\) of \(R\) such that \(K^2=0\) and \(K\oplus K^*\) is not a *-minimal ideal of \(R\). *-maximal ideals of a ring with involution are investigated and described in terms of maximal ideals, in particular, if \(R^2=R\) and \(M\) is a maximal ideal of \(R\), then \(M\cap M^*\) is a *-maximal ideal of \(R\). In a ring with involution every proper *-prime ideal is a *-maximal ideal if and only if every proper prime ideal is maximal.
scientific article; zbMATH DE number 1066093