Are rings closed under addition and multiplication? I've looked at a couple sources and neither of them state that rings are closed under addition and multiplication. For whatever reason, however, I believe that rings are defined to be closed under addition and multiplication. Am I correct? If so, why do authors not bother with this (in my opinion) crucial property?
 A: 99.999% of authors all include this property, at least implicitly.
Perhaps you are just not understanding where it appears implicitly.
"Closure" is implicit in the definition of a binary operation, so that is why most authors do not bother to mention it. It is not really important, it is just taken for granted that the reader understands the definition of binary operations on a set are necessarily "closed under the operation."
Addressing your sources:
Wikipedia:

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations ...

and then 

In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set.

(Sorry, I cannot seem to find the second book.)
While it is often used as an axiom for a binary operation on $X$ itself, it is useful to describe subsets of $X$ and their relation to the binary operation with "closure." Namely, a subset can be closed (or not) under the binary operation. That's why it becomes convenient to say things like "A subgroup $H$ of $G$ is a subset which is closed under the binary operation and inversion."
