Ring spectra and modules

Definition 8.1.6 (Ring spectra and modules). An associative ring spectrum is an associative algebra in \((\Sp ,\otimes ,\S )\), and a commutative ring spectrum is a commutative algebra in \(\Sp \). Every algebra \(A\) is canonically a left and right module over itself via its multiplication \(A\otimes A\to A\); this is called the regular \(A\)-module. We write \[ \Alg :=\Alg (\Sp ), \qquad \CAlg :=\CAlg (\Sp ) \] for their \(\infty \)-categories. For an associative ring spectrum \(R\), we abbreviate \[ \LMod _R:=\LMod _R(\Sp ), \qquad \RMod _R:=\RMod _R(\Sp ). \] If \(R\) is commutative, we write \(\Mod _R:=\Mod _R(\Sp )\) for its symmetric monoidal \(\infty \)-category of modules.