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  "body": "\n## Definition\n\nLet $R$ be a [[ring|0894J980RK]] with unit $1$.\nThe ring $R$ is *unit-regular* if for every $a \\in R$ there is a [[unit|019NFGFPR5]] $u$ of $R$ such that\n$$a u a = a.$$\n\n## Remarks\n\nA unit-regular ring is [[von Neumann regular|DEV24TSWQZ]].\nExample: a division ring $D$; for $a \\ne 0$ take $u = a^{-1}$, and for $a = 0$ take $u = 1$.\nExample: the matrix ring $M_n(F)$ over a field $F$.\nWrite $A = PEQ$ with $P, Q$ invertible and $E = \\operatorname{diag}(1, \\ldots, 1, 0, \\ldots, 0)$; then $u = Q^{-1} P^{-1}$ gives $AuA = PE^2Q = PEQ = A$.\nA unit-regular ring is Dedekind-finite: if $ab = 1$ and $a = aua$, then $1 = ab = auab = au$, so $a = u^{-1}$ and $ba = 1$.\nNon-example: let $V$ be a vector space with countable basis $e_1, e_2, \\ldots$ over a field, and $R = \\End(V)$.\nThe ring $R$ is von Neumann regular, but the shift $s(e_i) = e_{i+1}$ and $t$ with $t(e_1) = 0$, $t(e_{i+1}) = e_i$ satisfy $ts = 1 \\ne st$.\nSo $R$ is not Dedekind-finite, hence not unit-regular.\n",
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