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In the book "Lectures on Modules and Rings" by T.Y.Lam there are two definitions:

((8.2) Definition, T.Y.Lam.) We say that $N$ is a dense submodule of $M$ (written as $N\subseteq_d M$) if, for any $y\in M$ and $x\in M\setminus \{0\}, x\cdot y^{-1}N \neq 0$ (i.e., there exists $r\in R$ such that $xr\neq0$, and $yr\in N$). If $N\subseteq_d M$, we also say that $M$ is a rational extension of $N$.

((10.1) Definition, , T.Y.Lam) A ring $R'$ is said to be a right ring of fractions (with respect to $S\subset R$) if there is a given ring homomorphism $\phi: R \to R'$ such that:

(a) $\phi$ is $S$-inverting

(b) Every element of $R'$ has the form $\phi(a)\phi(s)^{-1}$ for some $a\in R$ and $s\in S$.

(c) $\ker (\phi) = \{r \in R : rs= 0 ~\text{for some}~ s\in S\}$.

I want to see how the two definitions are related. I have a feeling that they somehow refer to the same ring of quotients. Am I right? If yes, then how? If no, then why? What is the connection between these definitions?

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The condition of density is more general than rings of quotients: it describes the largeness of submodules in their supermodules. It is actually a stronger condition than that of being an essential submodule $N\subseteq_e M$. It isn't "a definition of a ring of quotients."

The connection is that condition $(b)$ implies $\phi(R)$ is a dense right $\phi(R)$ submodule of $R'$. This "largeness" condition is an important ingredient for classical rings of quotients. We are used to integral domains being dense in their field of fractions, but the strange thing is that there are noncommutative domains which can't be densely embedded in a division ring.

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