On some properties of extensions of commutative unital rings

We find necessary and sufficient conditions for the ring $R[\alpha]$ to be either a field or a domain whenever $R$ is a commutative ring with 1 and $\alpha$ is an algebraic element over $R$. This continues the studies started by Nachev (Compt. Rend. Acad. Bulg. Sci., 2004) and (Commun. Alg., 2005) as well as their generalization due to Mihovski (Compt. Rend. Acad. Bulg. Sci., 2005).