Let $A\subset R$ be rings such that every maximal ideal of $R$ contracts to a maximal ideal of $A$. Of course this is not always true, but it is the case for plenty of interesting examples, so it seems like this property might have a name. Does anyone know of one, or anywhere that this property is studied?
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2I cannot recall a name for this property. It happens in many situations, that do not have much in common: i) $A$ zero-dimensional (for instance a field), ii) $A \subset R$ integral, iii) $A,R$ of finite type over a field. – MooS May 15 '17 at 05:42