Morphism between projective schemes induced by injection of graded rings

Question

Let $A$ be a graded ring and $d>0$ be an integer. Define the graded ring $B$ such that $B_i=A_i$ if $d$ divides $i$ and $B_i=0$ otherwise. Is it true that a homomorphism of graded rings $B\hookrightarrow A$ induces isomorphism of schemes $\text{Proj}\,A\to\text{Proj}\,B$?