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 $\left{r\in R:rs=0\text{ for some }s\in S\right}$, 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.

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.

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 $\left\{r\in R:rs=0\text{ for some }s\in S\right\}$, 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.

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 $\left{r\in R:rs=0\text{ for some }s\in S\right}$, 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.