locally ringed space $(X,\mathcal{O}_X)$ isomorphic as Ringed Spaces to $Spec(A)$ but not isomorphic as Locally Ringed Spaces.

Question

I'm starting studying Hartshorne Chapter II and is the first time that I'm studying Schemes. I'm looking for some intuition viewing some examples.

I'm looking for an example of a locally ringed space $(X,\mathcal{O}_X)$ such that is isomorphic as Ringed space to the Spectrum of some ring $A$ but not isomorphic as Locally ringed spaces.

I think that some examples must exist but my intuition on ringed spaces is (at least right now) to vague. Thanks!

Answer 1

There is no such example: Any isomorphism of ringed spaces between two locally ringed spaces is already an isomorphism of locally ringed spaces.

This follows because a ring isomorphism $A\to B$ of local rings is a local isomorphism.