I have known that for a transitive model of ZFC, M, x is a Mgeneric filter of some forcing notion in M, then M[x] is least transitive model of ZFC which contains M and $\{x\}$. M[x] is just the set of all values of names in M. My question is, if I replace "a Mgeneric filter of some forcing notion in M" by an arbitrary subset of some element of M, how to define such a least model? i.e. M is a transitive model of ZFC, x is a set and x is a subset of some element of M. Can we define a least transitive model M[x] of ZFC such that $M\subset{M[x]}$ and $x\in{M[x]}$?
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