Natural dualities for quasivarieties generated by a finite commutative ring. (Q1771877)

Let \(R\) be a finite commutative ring with identity and \(J(R)\) its Jacobson radical. The authors show that the following assertions are equivalent: 1. \(R\) is \(\kappa \)-dualizable for some cardinal \(\kappa \) 2. \(R\) is strongly 4-dualizable 3. for all \(a,b \in J(R), \, a\cdot b =0\) (i.e.\ \(J(R)\) is self annihilating). Recall that \(R\) is \(\kappa \)-dualizable if it is dualized by the set of at most \(\kappa \)-ary operations and relations. The result is illustrated for the quasivariety of rings generated by \(\mathbb Z_{p^3}\) for a prime number \(p\).