<b>Motivation:</b> 

Say $F: D \to Sch$ is a diagram in the category of schemes, and we're interested in whether it has a [colimit][1] (gluings, pushouts, and "categorical" quotients are all examples of colimits).  Its colimit $Q$ in the category of locally ringed spaces always exists, and a scheme $Y$ is the scheme-colimit of $F$ iff there is a map $Q\to Y$ that is "initial among maps to schemes", i.e. any other map from $Q$ to a scheme factors through it.

Thus the problem of when colimits of schemes exists can be answered if we know when a locally ringed space admits a "schemification".

<b>Examples:</b> 

1) The process of turning a classical variety $V$ into a "variety with generic points" $V^s$  is a schemification.  Proving this boils down the classical affine case.  If $R$ is a finite-type reduced $k=\overline{k}$-algebra, and we map $V=mSpec(R)$ to a scheme $Y$, first cover $Y$ by affines $Spec(B_i)$, and then pull back this cover to $V$ and refine it by principal opens $mSpec(R_{f_{ij}})$, so we have maps $B_i \to R_{f_{ij}}$ that determine the map $V\to Y$ (by the adjunction of $Spec({\cal O}(-))$ to the inclusion $AffSch\hookrightarrow LRS)$.  But these define the desired map $V^s\to Y$ because $V^s$ can be obtained by gluing $Spec(R_{f_{ij}})$.

2) For $k=\overline{k}$, if we let $\mathbb{G_m}=\mathbb{A}^1_k\setminus 0$ act on $\mathbb{A}^1_k$, then the coequalizer in schemes of the action and the projection $G_m \times_k \mathbb{A}^1_k \rightrightarrows \mathbb{A}^1_k$ (i.e. the "categorical quotient") is $Spec(k)$, hence the locally ringed space coequalizer $Q$, which has two points and is not a scheme, has $Spec(k)$ as its schemification.

3) Some locally ringed spaces have no schemification; for example, two affine lines glued along their generic points.   This is precisely because the gluing diagram has no coequalizer in schemes, as per BCnrd and Anton Geraschenko's [answer here][2].


> When does a locally ringed space $X$ admit a "schemification", i.e. a map to a scheme $X^s$ through which any other map to a scheme must factor?

<b>EDIT:</b> (Response to Martin's answer)

4) Any locally ringed space $X$ with exactly one closed point $x$ has its "affinization", $X^a:=Spec({\cal O}_X(X))$, as a schemification.  This is because in a map $F:X\to Y$ with $Y$ a scheme, every point maps to a generization of $f(x)$, so and since open sets are closed under generization, $f$ factors through an affine neighborhood of $f(x)$.  But any map from $X$ to an affine factors uniquely through $X^a$, so we're done.  

5) Schemification commutes with disjoint unions, in that if $X= \coprod X_i$ then $X^s$ exists iff all $X_i^s$ exist, and in that case $X^s = \coprod X_i^s$. So *if $X$ is locally connected*, since it then decomposes into a disjoint union of connected clopen components, we might as well assume it is connected.

  [1]: http://en.wikipedia.org/wiki/Limit_(category_theory)#Colimits_2
  [2]: http://mathoverflow.net/questions/9961/colimits-of-schemes/23966#23966