Radicals of contracted graded rings (Q1902141)

Let \(S\) be a semigroup with a zero \(\theta\). By a contracted \(S\)-graded ring the author means a ring \(R = \bigoplus_{s \in S} R_s\) graded by \(S\) and such that \(R_\theta = 0\). The paper is concerned with the Jacobson, prime and locally nilpotent radicals of \(R\). Let \(\mathcal T\) denote one of these radicals. In the main result, necessary and sufficient structural conditions are found for \(S\) in order that \({\mathcal T} (R_e) = {\mathcal T} (R) \cap R_e\) for all contracted \(S\)-graded rings \(R\) and all \(e = e^2 \in S\). The motivation comes from the case where \(S\) is the semigroup of \(n \times n\) matrix units with zero.