Compatibility in a nearring (and its many variants) has established itself as an indispensible tool in the study of nearrings. Here the author studies compatible extensions of nearrings. Let \(R\) be a zero-symmetric near-ring with identity and let \(G\) be a faithful \(R\)-module. A subnearring \(S\) of \(M_0(G)\) is called a compatible extension of \(R\) relative to \(G\) if \(S\) contains \(R\), \(G\) is a compatible \(S\)-module and the \(R\)-ideals and \(S\)-ideals of \(G\) coincide. It is shown that the set of all such extensions is a complete lattice with respect to intersection and join; the latter is given by the subnearring generated by the union. It thus makes sense to talk about the compatible closure of a nearring which is then promptly characterized as the set of \(0\)-preserving polynomial maps of the extended group \(\langle G,+,R\rangle\). Further investigations involve a discussion of the least as well as the largest elements of this lattice.
The relations between the least element of this lattice and several types of centralizers and commutators of nearring modules are given; often involving the compatible closure of a nearring. Finally, the relation between the largest element of this lattice and the uniqueness of minimal factors is discussed and then applied to \(1\)-affine completeness of the \(R\)-module \(G\). Amongst others, a more general understanding of the notion of lonesomeness is given and used. Lonesomeness describes a certain type of interval in the lattice of \(CP_0(G)\)-ideals. Here \(CP_0(G)\) denotes the congruence preserving maps of \(M_0(G)\) for an expanded group \(G\).