I am reading a lemma on noetherian integral domains but I am stuck, I am bring it up here hoping for help. The original passage is in one big fat paragraph but I broke it down here for your easy reading. Thanks beforehand for all your time and help, let me know if I forget to include any underlying lemmas.
LEMMA: Let $M$ be an $R$-module. Let $T$ be maximal among the ideals of $R$ such that $M$ possesses a submodule $L$ for which $L/LT$ is not noetherian. Then $T$ is a prime ideal of $R$.
PROOF: (1) We are assuming that $M$ possesses a submodule $L$ for which $L/LT$ is not noetherian. Thus, as $L/LR = L/L$ is noetherian, $T \neq R$. [QUESTION: Here, I understand $LR=L$, but I am totally lost on how $L/LR=L/L$ is suddenly notherian.]
(2) Let us assume, by way of contradiction, that $T$ is not prime. Then $R$ possesses ideals $U$ and $V$ such that $T \subset U, T \subset V$ , and $UV \subseteq T$.
(3) The (maximal) choice of $T$ forces $L/LU$ and $LU/LUV$ to be noetherian. [QUESTION: I am lost on how the maximal choice of $T$ forces $L/LU$ and $LU/LUV$ to be noetherian.]
(4) Thus, by Lemma below, $L/LUV$ is noetherian. [QUESTION: Does it mean that since $L/LU$ and $LU/LUV$ are noetherian, therefore $L$, $LU$ and $LUV$ are noetherian, and therefore $L/LUV$ is noetherian?]
(5) On the other hand, as $UV \subseteq T, LUV \subseteq LT$. Thus, $L/LT$ is a factor module of $L/LUV$. [QUESTION: Here, I am begging explanation on how $L/LT$ is a factor module of $L/LUV$.]
(6) Thus, as $L/LUV$ is noetherian, $L/LT$ is noetherian; cf. Lemma below. This contradiction finishes the proof.
This is the lemma quoted above: Let $M$ be an $R$-module, and let $L$ be a submodule of $M$. Then $M$ is noetherian if and only if $L$ and $M/L$ are noetherian.