Should a ring be closed under multiplication?

In the definition of a ring, it is nowhere stated that it must be closed under multiplication. But it seems to be true for all the examples of rings that I've seen so far. So, is this implicitly assumed in the definition or can it be proved?

Solution 1:

A ring is an abelian group $R$ with an additional operation $\times$, that is, a function $\times:R\times R\to R$, satisfying the various axioms. The fact that this function has codomain $R$ is exactly the fact that $R$ is closed under multiplication.

Solution 2:

I suspect you are misreading the definition. The definition I know says that multiplication is a "binary operation" on the underlying set which means that it "$ab$" is a mapping from $X \times X\rightarrow X$. That is, the result of "$ab$", for $a$ and $b$ two members of the ring is again a member of the ring. I.e. it is closed under multiplication.