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In algebra, amodule spectrum is aspectrum with an action of aring spectrum; it generalizes amodule in abstract algebra.
The ∞-category of (say right) module spectra isstable; hence, it can be considered as either analog or generalization of thederived category of modules over a ring.
Lurie defines theK-theory of a ring spectrumR to be theK-theory of the ∞-category ofperfect modules overR (a perfect module being defined as a compact object in the ∞-category of module spectra.)
