Uniform Coherence in Amalgamated Algebra Along an Ideal
摘要
The concept of uniform coherence was first introduced by Soublin and has since been extensively explored by numerous researchers. A ring A is defined as uniformly coherent if there exists a function \(\phi : \mathbb {N} \rightarrow \mathbb {N}\) , where \(\mathbb {N}\) represents the set of natural numbers, such that for every \(n \in \mathbb {N}\) and each nonzero homomorphism \(f : A^n \rightarrow A\) , the kernel \(\ker (f)\) can be generated by \(\phi (n)\) elements. In this paper, we investigate how the property of uniform coherence is transferred to commutative ring extensions. We establish necessary and sufficient conditions for rings such as \(A\bowtie ^{f}J\) , \(A\bowtie I\) , and \(A\propto E\) to possess uniform coherence, covering various classes of ideals and A-modules. Our study contributes to the development of new classes of rings that meet this criterion, and we also introduce new families of rings that demonstrate the distinctiveness of the categories of Noetherian and uniformly coherent rings.