Isomorphism of ($\mathbb{Z}/{(n)}$-graded) Rings

Question

Let $A=\bigoplus_{d=0}^n A_g$ and $B=\bigoplus_{d=0}^n B_h$ be $\mathbb{Z}/{(n)}$-graded rings. In particular, we assume $A_n\ne 0$ and $B_n\ne 0$. Let $\phi:A\to B$ be an isomorphism of rings. My question is: Is $\phi(A_d)=B_d$? In other words, is $(\phi,\mathrm{id}_{\mathbb{Z}/{(n)}})$ an isomorphism of graded rings? If no, is there an automorphism $\chi$ of $(\mathbb{Z}/{(n)},+)$ such that $(\phi,\chi)$ is an isomorphism of graded rings?

Answer 1

Let $V$ be a vector space over a field $k$ and turn $R=k\oplus V$ into a ring so that the product of a pair of elements in $V$ is zero. Any grading of the vector space $R$ such that $1$ is homogeneous of degree zero is compatible with the ring structure. Picking gradings such that the homogeneous components are of different dimensions provides examples of graded rings which are isomorphic are rings but not as graded rings, not even up to automorphisms of the group.