Projective (or injective) object in a subcategory

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Let AA be an abelian category and BB be a full subcategory of AA. Suppose that BB is abelian and that the inclusion of BB in AA is exact. My question is: if an object PP of BB is projective in BB, then is it true that PP is projective in AA? (And what about the injective case?)

For example, consider a noetherian ring RR, take A=ModRA=ModR and B=modRB=modR (subcategory of finitely generated RR-modules). Then, if PP is a finitely generated RR-module which is projective in modRmodR, is it true that PP is a projective object in ModRModR? (An the injective case?)

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