Abstract
Let R ⊆ T be an extension of commutative rings with identity and H ( R , T ) (respectively, h ( R , T ) ) the composite Hurwitz series ring (respectively, composite Hurwitz polynomial ring). In this article, we study equivalent conditions for the rings H ( R , T ) and h ( R , T ) to be PF-rings and PP-rings. We also give some examples of PP-rings and PF-rings via the rings H ( R , T ) and h ( R , T ) .
Highlights
Let R ⊆ T be an extension of commutative rings with identity
We study equivalence conditions for composite Hurwitz rings H( R, T ) and h( R, T ) to be PF-rings and PP-rings, where R ⊆ T is an extension of commutative rings with identity
In [6], the authors introduced the notion of the composite Hurwitz rings and, in [7], they investigated further research
Summary
Let R be a commutative ring with identity and let H( R) be the set of formal expressions of the n type ∑∞. The ring H( R) is called the Hurwitz series ring over R. The Hurwitz polynomial ring h( R) is the subring of H( R) consisting of formal expressions of the form ∑in=0 ai X i. Let R ⊆ T be an extension of commutative rings with identity. The rings H( R, T ) and h( R, T ) are called the composite Hurwitz series ring and the composite Hurwitz polynomial ring, respectively. Let R ⊆ T be an extension of commutative rings with identity, u : R → T the natural monomorphism and v : H( T ) → T the canonical epimorphism. H( R, T ) can be understood as a pullback of R and H( T ) as follows: H( R, T ) = R × T H( T ) −−−−→. The readers can refer to [1,2,3,4] for the Hurwitz rings and to [5,6,7] for the composite Hurwitz rings
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