ON THE WEAK GLOBAL DIMENSION OF A SUBCLASS OF PRUFER NON-COHERENT RINGS
3
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
It is known that if R is a coherent Pr¨ufer ring, which is necessarily a Gaussian ring, then its weak global dimension w.gl.dim(R) must be 0, 1, or ∞. In this paper, we investigate the possible values of the weak global dimension for a broader class of Pr¨ufer rings that are not necessarily coherent. Our analysis employs four conceptually distinct proofs, each relying on different homological techniques, including localization at the nilradical, finitistic projective dimension, and flatness properties. The results extend the classical framework to a non-coherent setting by incorporating the effective HD framework, which serves as a surrogate for coherence in controlling homological dimensions. This work aims to deepen the understanding of the weak global dimension in the context of non-coherent Pr¨ufer rings and provide a unified perspective on its behavior.
Abstract
It is known that if R is a coherent Pr¨ufer ring, which is necessarily a Gaussian ring, then its weak global dimension w.gl.dim(R) must be 0, 1, or ∞. In this paper, we investigate the possible values of the weak global dimension for a broader class of Pr¨ufer rings that are not necessarily coherent. Our analysis employs four conceptually distinct proofs, each relying on different homological techniques, including localization at the nilradical, finitistic projective dimension, and flatness properties. The results extend the classical framework to a non-coherent setting by incorporating the effective HD framework, which serves as a surrogate for coherence in controlling homological dimensions. This work aims to deepen the understanding of the weak global dimension in the context of non-coherent Pr¨ufer rings and provide a unified perspective on its behavior.