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E infinity ring spectra were defined in 1972, but the term has since acquiredseveral alternative meanings. The same is true of several related terms. Thenew formulations are not always known to be equivalent to the old ones and evenwhen they are, the notion of "equivalence" needs discussion: Quillen equivalentcategories can be quite seriously inequivalent. Part of the confusion stemsfrom a gap in the modern resurgence of interest in E infinity structures. Einfinity ring spaces were also defined in 1972 and have never been redefined.They were central to the early applications and they tie in implicitly tomodern applications. We summarize the relationships between the old notions andvarious new ones, explaining what is and is not known. We take the opportunityto rework and modernize many of the early results. New proofs and perspectivesare sprinkled throughout.

What precisely are $E_{\infty}$ ring spaces and $E_{\infty}$ ring spectra?