Prime ideals of group graded semirings and their smash products. (Q1029937)

This paper generalizes some ring theoretic results of \textit{M. Cohen} and \textit{S. Montgomery} [Trans. Am. Math. Soc. 282, 237-258 (1984; Zbl 0533.16001)]. The main results are: 1. Let \(R\) be a yoked semiring graded by a finite group \(G\). If \(P\subsetneq Q\) are subtractive prime ideals of \(R\), then \(P\cap R_1\subsetneq Q\cap R_1\), where \(R_1\) is the identity component of \(R\). (Incompatibility). 2. Let \(R\) be a \(K\)-semialgebra graded by a finite group \(G\). i) If \(R\) is yoked and \(P\) is a subtractive prime ideal of \(R\), then there exists a prime \(Q\) of \(R\#K[G]^*\) such that \(Q\cap R=P_G\), \(Q\) is unique up to its \(G\)-orbit \(\{Q^g\}\) and \(P_G\#K[G]^*=\bigcap_{g\in G}Q^g\), a \(G\)-prime ideal of \(R\#K[G]^*\); ii) If \(I\) is any subtractive prime ideal of \(R\#K[G]^*\), then \(I\cap R=P_G\) for some subtractive prime \(P\) of \(R\). (Orbit Problem).