I am trying to compute all possible closed immersions of scheme $Spec(K)$, where $K$ is a field? Here is my try:
Suppose $f : X \rightarrow Spec(K)$, is closed immersion, then $X$ must be a point and hence $X = Spec(A)$, where $A$ has only one prime ideal. Since $f$ induce a surjective map on structure sheaf which implies that (along with closed immersion) that the induce map $K \rightarrow A$ is surjective, which implies that $A$ is $K$ module and has dimension as a $K$ module at most $1$. Since dimension can not be zero so $A$ as $K$ module must be isomorphic to $K$. So the only possible closed immersion of $Spec(K)$ is a map $Spec(K) \rightarrow Spec(K)$. Am I correct?