# Concept of a subring in Atiyah-Macdonald's book [duplicate] I think this definition is wrong, because nothing guarantees that the subring is closed to additive inverses.

Thanks

• I don't think I quite understand why you think his definition is wrong... Sep 24, 2013 at 14:54
• @RobertAuffarth: The definition seems to imply that $\mathbb N$ is a "subring" of $\mathbb Z$. Sep 24, 2013 at 14:57
• Oh! Now I understand, thanks. Sep 24, 2013 at 15:00
• Yep, seems like a problem. What edition of the book are you looking at? A "hack" fix would be to change it to: "and contains the additive inverse of the identity element of $A$." But better to just assert that $S$ is an additive subgroup. Sep 24, 2013 at 15:01
• @ThomasAndrews I think it's the first one from 1969. Sep 24, 2013 at 15:06

Yes, it is a typo. In Russian translation of the book it is written: "... if $S$ is an additive subgroup, closed under multiplication ..."
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.