(Edited)

Both notions seem to measure "largeness" of an ideal in a ring. how are they related? does one imply the other and vice versa?

i am reading this book "Hereditary Noetherian Prime Rings and Idealizers" by Levy and Robson.

On Page 58, Lemma 13.6, it says:

(Here $R$ is a Hereditary Noetherian Prime (HNP) ring)

All simple right $R$-modules occur as submodules of $R_\mbox{quo}/R$, where $R_\mbox{quo}$ is a ring of quotients of $R$. (indeed $R_\mbox{quo}$ is the injective hull $E(R)$ of $R$)

Let me copy the first few lines of the proof.

Let $X = R/M$ be a simple $R$-module. Since, by convention, $R \neq R_\mbox{quo}$, the socle of $R$ is zero. Therefore $M$ is essential in $R$. (and the proof goes on ...)

Here $M$ is maximal, and we have Maximal implies Essential. but at the same time it seems to suggest that under general conditions, Maximal does not necessarily imply Essential.

so my questions are

(i) when would one imply the other?

(ii) In those few lines i have copied, is the HNP property of $R$ involved?

(iii) As a side question, how important is it that a simple $R$-module can be embedded into the quotient module $E(R)/R$, where $E(R)$ is the injective hull of $R$? what is it really saying?