Let $p$ be a prime number, let $n$ be a positive integer. Let $B=\oplus_{d=0}^\infty B_d$ be a finitely generated graded $\mathbb{Z}/p^n\mathbb{Z}$-algebra. Suppose that there exist integers $N$ and $M$ such that $B\otimes \mathbb{F}_p$ is generated over $\mathbb{F}_p$ by elements of degree at most $N$ with relations in degree at most $M$. Is it true that $B$ is generated by its elements of degree at most $N$ with relations of degree at most $M$?
I think it should be true by Nakayama's lemma but I am not sure exactly how to apply it.
No. For instance, let $B=\mathbb{Z}/p^n\mathbb{Z}[x]/(px^2)$, where $x$ has degree $1$. Then $B\otimes\mathbb{F}_p\cong\mathbb{F}_p[x]$ is generated by elements of degree at most $1$ with relations of degree at most $-1$ (there are no relations at all). However, $B$ is not (assuming $n>1$), since it is not a free $\mathbb{Z}/p^n\mathbb{Z}$-algebra so any set of generators must have some relations.