• Title/Summary/Keyword: weak u-S-flat module

Search Result 3, Processing Time 0.015 seconds

Weak u-S-flat Modules and Dimensions

  • Refat Abdelmawla Khaled Assaad;Xiaolei Zhang
    • Kyungpook Mathematical Journal
    • /
    • v.63 no.3
    • /
    • pp.333-344
    • /
    • 2023
  • In this paper, we generalize the notions uniformly S-flat, briefly u-S-flat, modules and dimensions. We introduce and study the notions of weak u-S-flat modules. An R-module M is said to be weak u-S-flat if TorR1 (R/I, M) is u-S-torsion for any ideal I of R. This new class of modules will be used to characterize u-S-von Neumann regular rings. Hence, we introduce the weak u-S-flat dimensions of modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed.

THE w-WEAK GLOBAL DIMENSION OF COMMUTATIVE RINGS

  • WANG, FANGGUI;QIAO, LEI
    • Bulletin of the Korean Mathematical Society
    • /
    • v.52 no.4
    • /
    • pp.1327-1338
    • /
    • 2015
  • In this paper, we introduce and study the w-weak global dimension w-w.gl.dim(R) of a commutative ring R. As an application, it is shown that an integral domain R is a $Pr\ddot{u}fer$ v-multiplication domain if and only if w-w.gl.dim(R) ${\leq}1$. We also show that there is a large class of domains in which Hilbert's syzygy Theorem for the w-weak global dimension does not hold. Namely, we prove that if R is an integral domain (but not a field) for which the polynomial ring R[x] is w-coherent, then w-w.gl.dim(R[x]) = w-w.gl.dim(R).