Is a reducible variety projective if each of its irreducible component is?

Question

Suppose $X=X_1\cup\cdots\cup X_n$ is an irreducible decomposition of variety(scheme). Suppose each $X_i$ is projective, then is $X$ projective? Does this hold for proper case? The local ring at the intersection became unclear to me, I am not sure if the valuation criteria can still be verified at the intersection points? (I am reading the definition of a stable curve, which says a complete curve, or projective curve has some properties, I am not clear projectiveness means to treat each component or the whole can be embedded in some projective space?)

Answer 1

If each $X_i$ is proper, then $X$ is. (Just check the definition directly; you don't need to use the valuative criterion.)