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Let $R$ a ring with identity element, are there necessary conditions that grant that every left invertible element is also right invertible (equivalently if an element is left/ right invertible must be also invertible)? Is there a name for such rings?

Dietrich Burde
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Fabrizio
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  • Do you mean, if exist $x,y \in R$ such that $xy = 1$ and $yx \neq 1$? – Sewer Keeper Jun 05 '20 at 10:50
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    No, i mean if xy=1 then x and y are invertible. – Fabrizio Jun 05 '20 at 10:58
  • Here are a bunch of conditions which each imply Dedekind finiteness: 1) If the nilpotent elements form an ideal, 2) all idempotents are central (includes commutative rings, local rings and domains) 3) There are no infinite families of orthogonal idempotents (includes Noetherian+Artinian rings); 4) – rschwieb Jun 05 '20 at 12:25

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Such rings are called Dedekind-finite, i.e., the condition $ab=1$ implies $ba=1$.

Dietrich Burde
  • 140,055