# Question About Definition of Subring

In Dummit and Foote Abstract ALgebra, they define

A subring of the ring $$R$$ is a subgroup of $$R$$ that is closed under multiplication.

I know that since we check for closure and multiplication and subtraction, it is actually closed under addition, additive inverse and multiplication.

I don't understand how being closed under addition and additive inverses is in the defintion, does subgroup imply a subgroup under addition, even though they didn't specify the group law for the subgroup? Are subgroups implicitly groups under addition when talking about rings?

Yes, a subgroup of $$R$$ is indeed a subgroup of $$(R,+)$$, you are right. The reason is, that $$(R^{\times},\cdot)$$ is not a group (except for fields). Take $$R=\Bbb Z$$ the integers. It doesn't make sense to take about the group $$(\Bbb Z^{\times},\cdot)$$ under multiplication.