Is a defendant's admission of an enhancing allegation valid even if it does not include a specific admission of every factual element required to establish the enhancement?

Defendant acknowledges that a defendant's admission of an enhancing allegation is normally valid even if it does not include a specific admission of every factual element required to establish the enhancement. (People v. French (2008) 43 Cal.4th 36, 50.) However, he contends that this rule does not apply in his case because the trial court explicitly excluded any element of the enhancing allegation other than defendant's identity as the person who "committed those priors." Furthermore, when the court actually took defendant's admissions, it referred only to the first prior offense committed on October 21, 2008, and to the second prior offense committed on November 8, 2011, and omitted any mention of prison terms served for those offenses.