I am looking for various examples of non-regular rings whose every regular element is unit regular. In other words, a ring $R$, $R$ is not regular but for which if $a\in R$ is regular, then $a$ is unit regular.
Unit regular elements
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ring-theory
1 Answers
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Every local ring with nil radical is such a ring. Elements are split between nilpotent elements (which are obviously not regular) and units (which are obviously unit regular.) The link above points to several such rings.