Rafael ParraManuel Saorı́n2026-03-222026-03-22200910.48550/arxiv.0906.4357https://doi.org/10.48550/arxiv.0906.4357https://andeanlibrary.org/handle/123456789/83948Given a significative class $F$ of commutative rings, we study the precise conditions under which a commutative ring $R$ has an $F$-envelope. A full answer is obtained when $F$ is the class of fields, semisimple commutative rings or integral domains. When $F$ is the class of Noetherian rings, we give a full answer when the Krull dimension of $R$ is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.enKrull dimensionNoetherianMathematicsEnvelope (radar)Commutative ringClass (philosophy)Pure mathematicsCommutative propertyRing (chemistry)Local ringpreprint