Realising π^e _r–algebras by global ring spectra

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory, the global homotopy theory of Schwede (2018). Specifically, for a global ring spectrum R, we consider which classes of ring homomorphisms ηe W πe eR ! Se can be realised by a map ηW R ! S in the category of global R– modules, and what multiplicative structures can be placed on S. If ηe witnesses Se as a projective πe eR–module, then such an η exists as a map between homotopy commutative global R–algebras. If ηe is in addition étale or S0 is a Q–algebra, then η can be upgraded to a map of E1–global R–algebras or a map of G1–R–algebras, respectively. Various global spectra and E1–global ring spectra are then obtained from classical homotopy-theoretic and algebraic constructions, with a controllable global homotopy type.