Consider the expansion [Formula: see text] of a theory [Formula: see text] by a predicate for a submodel of a reduct [Formula: see text] of [Formula: see text]. We present a setup in which this expansion admits a model companion [Formula: see text]. We show that some of the nice features of the theory [Formula: see text] transfer to [Formula: see text]. In particular, we study conditions for which this expansion preserves the [Formula: see text]-ness, the simplicity or the stability of the starting theory [Formula: see text]. We give concrete examples of new [Formula: see text] not simple theories obtained by this process, among them the expansion of a perfect [Formula: see text]-free PAC field of positive characteristic by generic additive subgroups, and the expansion of an algebraically closed field of any characteristic by a generic multiplicative subgroup.

We show that if Z is a dp-minimal expansion of (Z, +, 0, 1) that defines an infinite subset of N, then Z is interdefinable with (Z, +, 0, 1, <). As a corollary, we show the same for dp-minimal expansions of (Z, +, 0, 1) which do not eliminate ∃ ∞ .

We study expansions of a vector space V over a field F, possibly with extra structure, with a generic submodule over a subring of F. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP 1 , NTP 2 and NSOP 1 . We also study induced independence relations in the expansion.

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