Root extension of generalized power series rings

Abstract

Let R 1 ⊆ R 2 be commutative rings with unity and let S 1 ⊆ S 2 be torsion-free abelian monoids. We give a characterization of when the extension R 1 [ S 1 ] ⊆ R 2 [ S 2 ] of monoid rings is a root extension. For torsion-free, ≤ -cancellative, strictly ordered abelian monoids ( S 1 , ≤ ) ⊆ ( S 2 , ≤ ) , we also give a characterization of when the extension [ ​ [ R 1 S 1 , ≤ ] ​ ] ⊆ [ ​ [ R 2 S 2 , ≤ ] ​ ] of generalized power series rings is a root extension.