Let $R$ be a commutative ring. If $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of modules, we have that $\operatorname{Supp}M=\operatorname{Supp}M'\cup \operatorname{Supp}M''$. Furthermore can we obtain that $\dim M=\dim M'+\dim M''$? In fact, I want to know that given a module $M$ with $\dim M=n$, is it possible to construct an exact sequence as above with $\dim M'=0$ and $\dim M''=n-1$? 



An elementary question about the Krull dimension of modules
The dimension of $M$ is the maximum of the dimensions of $M'$ and $M''$, so what you are asking for cannot happen (at least not in the noetherian case). To Dylan: the dimension of a module is the dimension of its support.


