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[parent] inverses in rings (Topic)

Let $ R$ be a ring with unity $ 1$ and $ r \in R$. Then $ r$ is left invertible if there exists $ q \in R$ with $ qr=1$; $ q$ is a left inverse of $ r$. Similarly, $ r$ is right invertible if there exists $ s \in R$ with $ rs=1$; $ s$ is a right inverse of $ r$.

Note that, if $ r$ is left invertible, it may not have a unique left inverse, and similarly for right invertible elements. On the other hand, if $ r$ is left invertible and right invertible, then it has exactly one left inverse and one right inverse. Moreover, these two inverses are equal, and $ r$ is a unit.



"inverses in rings" is owned by Wkbj79.
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See Also: Klein 4-ring, left and right unity of ring

Also defines:  left invertible, right invertible, left inverse, right inverse

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Cross-references: unit, ring with unity
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This is version 1 of inverses in rings, born on 2007-05-24.
Object id is 9457, canonical name is InversesInRings.
Accessed 1476 times total.

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AMS MSC: 16-00 (Associative rings and algebras :: General reference works )
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