Graded \(P\)-radical (Q2732371)

Let \(M\) be a monoid. An \(M\)-graded ring \(R=\bigoplus(R_m:m\in M)\) is said to be a graded \(M\)-ring, if for each \(a\in R\) there exists a polynomial \(f(x)\) over the integers with zero constant term such that \(f(a)=0\). The class of all \(M\)-graded rings \(P_G\) is a graded radical class in the sense of Kurosh and Amitsur. The class of \(P_G\)-semisimple \(M\)-graded prime rings is a graded special class, whence \(P_G\) is a graded special radical. For \(M\)-graded \(R\)-modules \(V\), \(P_G\)-modules are defined, and it is proved that \(P_G=\bigcap(0:V)\), where \(V\) ranges over all \(R\)-\(P_G\)-modules.