On the radicals of semigroup rings

The author investigates properties of the radicals of semigroup rings, that can be considered as rings, graded by a band. A characterization of the Brown–McCoy radical of a band-graded ring is obtained. For a semigroup ring of a semilattice it is proved that $\rho(R[\Omega])\subseteq\rho(R)[\Omega]$ holds for any radical $\rho$ (in the sense of Kurosh–Amitsur). The technique of this paper is developed from the one used by W. D. Munn in [3].