Non-triviality conditions for integer-valued polynomial rings on algebras
G Peruginelli, NJ Werner - Monatshefte für Mathematik, 2017 - Springer
Monatshefte für Mathematik, 2017•Springer
Let D be a commutative domain with field of fractions K and let A be a torsion-free D-algebra
such that A ∩ K= DA∩ K= D. The ring of integer-valued polynomials on A with coefficients in
K is Int _K (A)={f ∈ KX ∣ f (A) ⊆ A\} I nt K (A)= f∈ KX∣ f (A)⊆ A, which generalizes the
classic ring Int (D)={f ∈ KX ∣ f (D) ⊆ D\} I nt (D)= f∈ KX∣ f (D)⊆ D of integer-valued
polynomials on D. The condition on A ∩ KA∩ K implies that DX ⊆ Int _K (A) ⊆ Int (D) DX⊆
I nt K (A)⊆ I nt (D), and we say that Int _K (A) I nt K (A) is nontrivial if Int _K (A) ≠ DXI nt K …
such that A ∩ K= DA∩ K= D. The ring of integer-valued polynomials on A with coefficients in
K is Int _K (A)={f ∈ KX ∣ f (A) ⊆ A\} I nt K (A)= f∈ KX∣ f (A)⊆ A, which generalizes the
classic ring Int (D)={f ∈ KX ∣ f (D) ⊆ D\} I nt (D)= f∈ KX∣ f (D)⊆ D of integer-valued
polynomials on D. The condition on A ∩ KA∩ K implies that DX ⊆ Int _K (A) ⊆ Int (D) DX⊆
I nt K (A)⊆ I nt (D), and we say that Int _K (A) I nt K (A) is nontrivial if Int _K (A) ≠ DXI nt K …
Abstract
Let D be a commutative domain with field of fractions K and let A be a torsion-free D-algebra such that . The ring of integer-valued polynomials on A with coefficients in K is , which generalizes the classic ring of integer-valued polynomials on D. The condition on implies that , and we say that is nontrivial if . For any integral domain D, we prove that if A is finitely generated as a D-module, then is nontrivial if and only if is nontrivial. When A is not necessarily finitely generated but D is Dedekind, we provide necessary and sufficient conditions for to be nontrivial. These conditions also allow us to prove that, for D Dedekind, the domain has Krull dimension 2.
Springer
以上显示的是最相近的搜索结果。 查看全部搜索结果