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  "body": "\n## Summary\n\nA ring $R$ is unit-regular if every element $a$ has a unit $u$ with $aua = a$.\nThis strengthens [[von Neumann regularity|DXX7GNY3HE]], where $u$ is only required to be some element of $R$.\nUnit-regular rings are [[Dedekind-finite|GV1HKEAWVN]], and for a vector space $V$ the ring of endomorphisms is unit-regular exactly when $V$ is finite-dimensional.\nThe notion matters in the study of cancellation of modules and direct sum decompositions.\n\n## Explanations\n\n### L1 human\n\nIn a von Neumann regular ring each element $a$ has a \"partial inverse\" $x$ with $axa = a$.\nIn a unit-regular ring the partial inverse can be chosen to be a genuinely invertible element.\nFor a square matrix $A$ over a field, one can write $A = PEQ$ with $P, Q$ invertible and $E$ diagonal with entries $0$ and $1$, and then $u = Q^{-1}P^{-1}$ works.\nFor infinite-dimensional spaces this fails: the shift operator that moves every basis vector one step forward has no such invertible partner.\n",
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