First, we characterize separable (resp: H-separable) ring extensions by properties on the endomorphism ring

viewed as an

-bimodule. On the second hand, using results on separability, we check conditions on a ring extension for which, conoetherian or coartinian property on B are transferred on A and conversely. We prove that, if is an H-separable ring extension such that then B is conoetherian (resp: coartinian) if and only if A is conoetherian (resp: coartinian).

conoetherian, coartinian, H-separable (separable) extension, Morita equivalence, injective, projective, generator, normalizing, centralizing.