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In his excellent monograph "Lectures on Modules and Rings" (GTM 189), T.Y. Lam remarks (10.13), p. 302, that there is no direct method of computing the kernel of the (co-)unit of the adjunction when localizing a ring $R$ w.r.t. a right denominator set $S$. My question then: Is there really no way of identifying $\text{ker}\,\varepsilon$ "from scratch", i.e. as $\lbrace r\in R \;:\; rs=0$ for some $s\in S\rbrace$, just from the desired universal property (but still assuming that $S$ is a right denominator set) ? Any help/insightful comments would be dearly appreciated !

Kind regards and thank you in advance, St.

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    $\begingroup$ I've asked the same question in the commutative setting here mathoverflow.net/questions/86923 $\endgroup$
    – Martin Brandenburg
    Commented May 21, 2012 at 13:13
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    $\begingroup$ @Martin: the commutative setting is really different!!! The common point of view for localizing a module $M$ on a commutative ring $R$ with respect to a multiplicative subset $\Sigma\subseteq R$ is to view the elements of $\Sigma$ as endomorphisms of $M$ and so take the direct limit of $|\Sigma|$-many copies of $M$ with these transition maps. When $R$ is not commutative you cannot do so if $\Sigma$ is not included in the center $Z(\Sigma)$. This is a non-trivial complication (in concrete situations it makes really a lot of difference, I can give you concrete examples)! $\endgroup$
    – Simone Virili
    Commented Dec 2, 2012 at 17:55

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