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Let $R$ be a (not necessarily commutative) ring. Let $\tau$ be an automorphism of $R$. Consider the localisation of $R$ at a set of multiplicative elements which satisfy the ore condition, say $X$. When does $\tau$ extend to $RX^{-1}$?

My thoughts: Since any element of $RX^{-1}$ is of the form $rx^{-1}$, where $r\in R$ and $x\in X$, we are really asking: When is $\tau(x)^{-1}$ defined? This is definitely defined if $X$ is invariant under $\tau$ but is this it? What if $\tau(x)=ux$ where $u$ is a unit in $R$? This seems fine too. But here $X$ is not left invariant. Can anyone help me with the necessary and sufficient conditions?

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  • $\begingroup$ Are you assuming the right Ore condition, or some two-sided Ore condition? Are you assuming that $X$ consists of regular elements, or are you allowing zero-divisors? Finally, what have you done to try to approach this problem? $\endgroup$
    – Pace Nielsen
    Commented Oct 25, 2016 at 23:26
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    $\begingroup$ The universal property of localization answers the question. $\endgroup$
    – HeinrichD
    Commented Oct 26, 2016 at 6:17

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