Definition 8.5.2 (Projective module). Let \(R\) be a connective associative ring spectrum. A left \(R\)-module \(P\) is projective if it is a t-projective object of \(\LMod _R\) in the sense of Definition 6.4.1; that is, if the mapping spectrum functor \(\hom _R(P,-)\colon \LMod _R \to \Sp \) is t-exact.
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