Suppose we are given two graded (commutative) rings $A$ and $B$ and a graded homomorphism $\psi:A\rightarrow{B}$ between them. Suppose moreover that $\psi$ is surjective in each degree i.e. that $\psi(A_n)=B_n$ for all $n$. If $\psi$ is injective in degrees $0$ and $1$, must it also be injective everywhere else? (all relations in $A$ come from these two degrees). If not in general, does it make a difference if we assume $B$ is an integral domain? (or even more specifically if it is a polynomial ring with coefficients in an integral domain).
Surjective graded homomorphism of rings also an isomorphism?
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abstract-algebra
ring-theory