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I was going through the minimum variance unbiased estimators and I am confused about the concept of MVUE and UMVUE. Is the unbiased estimator whose variance attaining CRLB a UMVUE or MVUE? I referred two books. One of those(A first course on parametric inference by B.K.Kale ) says that the unbiased estimator attaining CRLB is MVUE and the other(An introduction to probability and statistics by Rohatgi) says that it is UMVUE. So, are they same?

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MVUE and UMVUE are two different names to the same concept: unbiased estimators that achieve lowest variance among all other unbiased estimators, uniformly in all possible parameters. Consequently, an unbiased estimator that attains Cramer Rao lower bound is MVUE/UMVUE.

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