ag.algebraic geometry - ring-valued points of locally ringed spaces

of course, one should expect that the concept of ring-valued points is not well-behaved for locally ringed spaces (LRS). I want to see examples for this.

so consider $LRS to Set^{Ring}, X mapsto X(-)=Hom(Spec - , X)$. if $A$ is a local ring, whose maximal ideal is principal, and $hat{A}$ its completion, and we regard local rings as locally ringed spaces whose underlying set is just one point, then $A to hat{A}$ induces a bijection $Hom(Spec R,A) to Hom(Spec R,hat{A})$ (I'll add the proof if you want). this shows that the functor is not full. but how can we see that it is not faithful?

For example, for local rings $A$, we have

$Hom_{LRS}(Spec R,A)={phi in Hom_{Ring}(A,R) : phi(mathfrak{m}_A) subseteq rad(R)}$.

If $f,g$ are local homomorphisms inducing the same maps $Hom_{LRS}(Spec -,B) to Hom_{LRS}(Spec -,A)$, it seems that they don't have to be identical ...

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