, Volume 16, Issue 6, pp 1647-1660
Date: 26 Sep 2012

Zaks’ Lemma for Coherent Rings

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Abstract

Let A be a left and right coherent ring and C A (resp., \(C_{A^{\mathrm{op}}}\) ) a minimal cogenerator for right (resp., left) A-modules. We show that \(\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}}\) whenever flat dim C A  < ∞ and \(\mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty\) , and that \(\mathrm{flat \ dim \ }C_{A} = \mathrm{flat \ dim \ }C_{A^{\mathrm{op}}} < \infty\) if and only if the finitely presented right A-modules have bounded Gorenstein dimension.

Presented by Kenneth Goodearl.