I have seen the concept of length being applied to rings. What is exactly mean by that? What does length mean in a statement like "the composition length of RR is 2, but the composition length of RR is n+1>2."? Try to explain it in a simple fashion, please.
Thanks in advance
Let $R$ be a ring. The length of a left (or right) $R$-module $M$ is the maximum length $n$ of a chain of left (or right) $R$-modules $$ M_0 \subsetneq M_1 \subsetneq \dotsb \subsetneq M_n = M $$ if it exists, otherwise it is said to be infinite. The length of $R$ is its length as a left (or right) $R$-module.
Note that the only module of length $0$ is the zero module, because it is contained in every other module. Furthermore, it can be shown that a module (or ring) has finite length if and only if it is both Noetherian and Artinian.
This implies that this notion of length is of limited use for commutative rings, because a commutative Noetherian ring is Artinian if and only if all of its prime ideals are maximal. In particular, every integral domain that is not a field has infinite length, because its $0$ ideal is prime. For example, if $n \in \Bbb{Z}$ is different from $0$ or $\pm 1$, then $$ \Bbb{Z} \supsetneq (n) \supsetneq (n^2) \supsetneq (n^3) \supsetneq \dotsb $$
Finally, note that the lengths of a ring seen as a left or right module can differ, even if they are both finite. You can find an example in this answer (that you linked to already).