On the description of radicals of semigroup rings of commutative semigroups (Q1909847)

A class \(\mathcal K\) of associative rings is called local if for any ring \(A\) having a local system \((A_i\mid i\in I)\) of subrings \(A_i\in {\mathcal K}\) it follows \(A\in {\mathcal K}\). A ring \(R\) is called \(\Omega\)-semilattice of its subrings \(R_\alpha\) \((\alpha\in\Omega\), \(\Omega\) a semilattice) if \(R(+)=\bigoplus_\alpha R_\alpha (+)\) and \(R_\alpha R_\beta\subset R_{\alpha\beta}\) holds for all \(\alpha,\beta\in\Omega\); the subrings \(R_\alpha\) are called components of \(R\). Suppose that every component \(R_\alpha\) has the unit element \(e_\alpha\) and for given \(\tau\in\Omega\), \(r\in R_\tau\) and a finite (possibly, empty) set \(\Lambda\) of elements in \(\tau\Omega\) take the product \((r;\tau,\Lambda)\dot=r\cdot\prod_{\lambda\in\Lambda}(e_\tau-e_\lambda)\); it is defined \((r;\tau,\emptyset)=r\). The main result of this paper is the following (Theor. 1): If \(\rho\) is a hereditary radical in the sense of Kurosh and Amitsur with its radical class being local and \(R\) is a semilattice of rings having unit elements then the radical \(\rho(R)\) coincides with the additive subgroup \(\rho'(R)\leq R(+)\) generated by the set \({\mathcal P}(R)\) of elements \((r;\tau,\Lambda)\) such that either for any \(\alpha\leq\tau\) it holds \(e_\alpha r\in\rho(R_\alpha)\), or there exists \(\lambda\in\Lambda\), \(\lambda\geq\alpha\). In\S 2 it is shown how to use this result for a reduction of the Jacobson radical \(J(K[S])\) of the semigroup ring of a commutative semigroup \(S\) to some ideals defined via polynomial rings; in this connection, see \textit{S. Amitsur} [Can. J. Math. 8, 355-361 (1956; Zbl 0072.02404)], also \textit{E. Jespers} and \textit{P. Wauters} [Proc. Int. Conf. Johannesburg 1985, North-Holland Math. Stud. 126, 43-89 (1986; Zbl 0599.20103)].