发表机构
School of Mathematics and Statistics, Tianshui Normal University（天水师范学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了$\phi$-Prüfer环的$\phi$-弱整体维数仅可能为0、1或无穷，并证明了$\phi$-Dedekind环的$\phi$-整体维数三分法，通过典范核和拼接定理建立了强$\phi$-环的维数等式。
AI 中文摘要
设$R$为一个$\phi$-环，并令$N=Nil(R)$。我们确定了每个$\phi$-Prüfer环的$\phi$-弱整体维数。其可能值仅为$0$、$1$和$\infty$：当$R/N$是域时值为$0$，当$R$是强环且$R/N$是非域Prüfer整环时值为$1$，当$R$不是强环时值为无穷。对$N$不需要幂零性假设。证明基于典范核$K=\ker(R\to\phi(R))$。在局部非强情形下，$K$是赋值整环$R/N$上的非零挠可除模，且其自Tor群满足\\[ Tor^R_{2n}(K,K)=0\quad(n\geq 1), \qquad Tor^R_{2n+1}(K,K)\neq 0\quad(n\geq 0). \\] 由此可得$fd_RK=\infty$，并且一个滤过余极限论证将此障碍转移到由非幂零主理想生成的循环商上。我们还证明了$\phi$-Dedekind环的$\phi$-整体维数的平行三分法。对于强$\phi$-环，我们在不假设商映射$R\to R/N$分裂的情况下获得了环变换定理。若$D=R/N$，$Q=Frac(D)$且$T=T(R)$为全商环，则\\[ R\cong T\times_Q D. \\] 利用Ferrand的平坦拼接定理和Milnor的投射拼接定理，我们证明了每个挠$D$-模$M$（通过$R\to D$视为$R$-模）满足\\[ fd_R(M)=fd_D(M), \qquad pd_R(M)=pd_D(M). \\] 因此，对于每个强$\phi$-环，\\[\phi-wgd(R)=wgld(R/N), \qquad \phi-gld(R)=gld(R/N). \\]
英文摘要
Let $R$ be a $ϕ$-ring and put $N=Nil(R)$. We determine the $ϕ$-weak global dimension of every $ϕ$-Prüfer ring. Its only possible values are $0$, $1$, and $\infty$: the value is $0$ when $R/N$ is a field, it is $1$ when $R$ is strong and $R/N$ is a nonfield Prüfer domain, and it is infinite when $R$ is not strong. No nilpotence assumption on $N$ is needed. The proof is based on the canonical kernel $K=ker(R\toϕ(R))$. In the local non-strong case, $K$ is a nonzero torsion divisible module over the valuation domain $R/N$, and its self-Tor groups satisfy \[ Tor^R_{2n}(K,K)=0\quad(n\geq 1), \qquad Tor^R_{2n+1}(K,K)\neq 0\quad(n\geq 0). \] It follows that $fd_RK=\infty$, and a filtered-colimit argument transfers this obstruction to cyclic quotients by nonnil principal ideals. We also prove the parallel trichotomy for the $ϕ$-global dimension of $ϕ$-Dedekind rings. For strong $ϕ$-rings we obtain a change-of-rings theorem without assuming that the quotient map $R\to R/N$ splits. If $D=R/N$, $Q=Frac(D)$ and $T=T(R)$ is the total quotient ring, then \[ R\cong T\times_Q D. \] Using Ferrand's flat patching theorem and Milnor's projective patching theorem, we prove that every torsion $D$-module $M$, regarded as an $R$-module through $R\to D$, satisfies \[ fd_R(M)=fd_D(M), \qquad pd_R(M)=pd_D(M). \] Consequently, for every strong $ϕ$-ring, \[ϕ-wgd(R)=wgld(R/N), \qquad ϕ-gld(R)=gld(R/N). \]
