Extensions and Pullbacks in QB-rings

We prove a new extension result for $QB-$rings that allows us to examine extensions of rings where the ideal is purely infinite and simple. We then use this result to explore various constructions that provide new examples of $QB-$rings. More concretely, we show that a surjective pullback of two $QB-$rings is usually again a $QB-$ring. Specializing to the case of an extension of a semi-prime ideal $I$ of a unital ring $R$, the pullback setting leads naturally to the study of rings whose multiplier rings are $QB-$rings. For a wide class of regular rings, we give necessary and sufficient conditions for their multiplier rings to be $QB-$rings. Our analysis is based on the study of extensions and the use of non-stable $K-$theoretical techniques.