Free modules over hjelmslev rings in which not every maximal linearly independent subset is a basis

First it will be shown that every left-noetherian AH-ring is left-artinian. For an AH-ring R every finite linearly independent subset of a free left R-module V can be completed to a basis of V. But a maximal linearly independent subset of a free left R-module V need not be a basis of V. For an H-ring R, every maximal linearly independent subset of a free left R-module V is a basis of V if and only if the H-ring R is left-noetherian or V is finitely generated.