# Does a subring of $\mathbb{Z}$ need to be closed under multiplication?

I know that the ring $\mathbb{Z}$ has the binary operation under addition, and when we are trying to identify whether a given ring is a subring of $\mathbb{Z}$, the subring must contain: the identity, inverse, and be closed under addition. My question is that does the subring also have to be closed under multiplication because multiplication is not the binary operation on $\mathbb{Z}$?

• possible duplicate of How to check a set of ring is a subring? – Dietrich Burde Dec 18 '14 at 21:34
• @DietrichBurde No this isn't a duplicate (of that question anyway). The subring test for this particular ring can be weakened. – Bill Cook Dec 18 '14 at 22:08

Why? Because multiplication in $\mathbb{Z}$ is merely repeated addition/subtraction. Since you already have closure under addition and additive inverses, closure under multiplication follows for free.
• Actually $\mathbb{Z}$ has no non-trivial subrings (in the usual terminology, where rings have a multiplicative unit). As you say, any additive subgroup of $\mathbb{Z}$ is an ideal and hence a subrng (where rngs don't necessarily have the multiplicative unit). – Rob Arthan Dec 18 '14 at 22:40