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We prove that if a local ring admits a (pd-)test module of finite completeintersection dimension, then it is a complete intersection ring. This answers,positively, a question proposed by Celikbas, Dao and Takahashi. To this aim, wefirst investigate another question raised by Celikbas and Sather- Wagstaffconcerning ascent properties of (pd-)test complexes under weakly regularhomomorphisms.

Test modules, weakly regular homomorphisms and complete intersection dimension