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To add to Achim's answer, Bousfield and Kan, "The core of a ring" JPAA 1972 and (correction) 1973 define a ring $R$ to be a "solid" if the multiplication map $R\otimes_{\mathbb{Z}}R\to R$ is an isomorphism. They then go on to characterise solid rings. There is also a related discussion on page 44 of their book, "Homotopy limits, completions and localizations".