Let $R$ be a ring and $M$ be an $R$-module. Also assume that $N$ is a complement in $M$ (recall that a submodule $K$ of $M$ is called a complement in $M$ if there exists a submodule $L$ of $M$ such that $K\cap L=\{ 0\}$ and $M=K+L$. In other words, $M=K\oplus L$).
Assume that $M$ is a module. Is there a non-negative integer, say $n(M)$, with the following conditions:
(1) $n(M)<\infty$;
(2) if $M\cong N$, then $n(M)=n(N)$;
(3) if $K$ is a proper complement in $M$, then $n(K)\lneq n(M)$.
In particular, I'm interested in the class of Noetherian moduels, but I don't know this class of modules is countable.
Can $n(G)$ be the number of direct summands of $M$? Can $n(G)$ be the uniform (or Goldie) dimension of $M$?
My try: I define $n(G)$ inductively: $n(\{0\})=0$ and $$n(M)=1+\sup\{ n(N): N\; \text{is a proper complement in}\; M\}.$$ But my problem is that whether $\sup\{ n(N): N\; \text{is a proper complement in}\; M\}$ exist or not.