Concept of a subring in Atiyah-Macdonald's book 
I think this definition is wrong, because nothing guarantees that the subring is closed to additive inverses.
Thanks
 A: I cannot believe that the definition of a subring in Atiyah-Macdonald is wrong or incomplete. I still try to find a way to read the definition so that it becomes true ;-).
By the way, instead of adding "if $S$ is an additive subgroup ...", it suffices to demand that $-1 \in S$. Because then we have $-x = (-1) \cdot x \in S$ for all $x \in S$.
By the way, even then this won't be the "correct" definition: A subring is not just a subset with properties. It is a subset with extra structure, namely a ring, and properties. A quite abstract definition would be: A subring of $R$ is a subobject of $R$ in the category of rings. Equivalently, and more explicit: A subring $S$ of $R$ is a ring (!) together with a homomorphism of rings $S \to R$, which is just an inclusion on the underlying sets. But in practice these are often just injective maps, not inclusions in the sense of set theory (for example, $\mathbb{Z} \to \mathbb{Z}[T]$). The category-theoretic definition might be abstract, but it is really useful in practice. Even abstract algebra texts such as the one by Atiyah-Macdonald use it without any explanation. So their definition of a subring is not just wrong because the closure under additive inverses is lacking, but also because this is not the definition which is used in the book.
A: Yes, it is a typo. In Russian translation of the book it is written: "... if $S$ is an additive subgroup, closed under multiplication ..."
