$\operatorname{Spec} \mathbb Z[x]$ is a group scheme. Then there is a morphism of schemes: $\operatorname{Spec} \mathbb Z[x]\times\operatorname{Spec} \mathbb Z[x]\rightarrow\operatorname{Spec} \mathbb Z[x]$ induced by the coaddition $a:\mathbb Z[x]\rightarrow \mathbb Z[x] \otimes\mathbb Z[x]$ mapping $x$ to $1\otimes x+x\otimes 1$. Now, the coaddition is not a morphism of rings, since it does not respect multiplication: $a(x^2)=(1\otimes x^2+x^2\otimes 1)\neq(1\otimes x + x\otimes 1)^2=a(x)\cdot a(x)$.
But $a$ should be a morphism of rings, since the addition on $\operatorname{Spec} \mathbb Z[x]$ is a morphism of affine schemes.