Radicals of semigroup rings (Q2564838)

A ring means an associative ring. Let \(P\) be a radical property, let \(P(R)\) be the \(P\)-radical of a ring \(R\). If \(S\) is a semigroup, then \(R[S]\) denotes the semigroup ring of \(S\) over a ring \(R\). The author gives a \(P\)-semisimplicity criterion for a semigroup ring \(R[S]\) in terms of congruences on a semigroup \(S\). The main result is the following one. For \(S\) a semigroup and \(R\) a ring, \(P(R[S])=0\) if and only if there exists a family \(\rho_\lambda\) (\(\lambda\in\Lambda\)) of congruences on \(S\) such that: (i) \(P(R[S/\rho_\lambda])=0\) for all \(\lambda\in\Lambda\); (ii) for any \(i,j\in\Lambda\) there exists \(k\in\Lambda\) such that \(\rho_i\cap\rho_j\supseteq\rho_k\); (iii) the intersection of all \(\rho_\lambda\) (\(\lambda\in\Lambda\)) is the equality relation on \(S\) (Theorem 3.2). The author applies his criterion for group rings (Corollary 3.3) and to the case all \(\rho_\lambda\) are Rees congruences (Theorem 3.4). For \(P\) a hereditary radical property, a criterion is given for a semigroup ring to be a \(P\)-radical ring.