Let $0\rightarrow N \rightarrow M\rightarrow P \rightarrow 0$ be an exact sequence of finitely generated $\mathbb{N}$ graded module over a commutative ring $R$.
The vanishing degree of a $\mathbb{N}$ graded module $M$ is defined to be the maximal number $m$ such that $M_m \neq 0$ and denoted by $v(M)$.
I guess that : $v(M)=\text{max}\lbrace v(N),v(P)\rbrace$ but I could not prove it. Could you please help me ? If I am wrong please show me a counter example.
Thanks.