Homological dimensions and special base changes

We relate the homological behavior of an associative ring R and those of the rings R/xR and Rx when x is a regular central element in R. For left weak global dimensions we prove wgldim(R)≤max⁡{1+wgldim(R/xR),wgldim(Rx)} with equality if wgldim(R/xR) is finite. The key point is a formula for flat dimensions of R-modules: fdRM=max⁡{fdR/xR((R/xR)⊗RLM),fdRxMx}. For left noetherian R we recover results of Li, Van den Bergh and Van Oystaeyen [3] on global and projective dimensions. Similar formulae hold for injective dimensions.