I think this definition is wrong, because nothing guarantees that the subring is closed to additive inverses.
Thanks
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asked Sep 24, 2013 at 14:51
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2$\begingroup$ I don't think I quite understand why you think his definition is wrong... $\endgroup$– rfauffarSep 24, 2013 at 14:54
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7$\begingroup$ @RobertAuffarth: The definition seems to imply that $\mathbb N$ is a "subring" of $\mathbb Z$. $\endgroup$– hmakholm left over MonicaSep 24, 2013 at 14:57
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1$\begingroup$ Oh! Now I understand, thanks. $\endgroup$– rfauffarSep 24, 2013 at 15:00
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$\begingroup$ 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. $\endgroup$– Thomas AndrewsSep 24, 2013 at 15:01
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$\begingroup$ @ThomasAndrews I think it's the first one from 1969. $\endgroup$– user42912Sep 24, 2013 at 15:06
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2 Answers
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Yes, it is a typo. In Russian translation of the book it is written: "... if $S$ is an additive subgroup, closed under multiplication ..."
answered Sep 24, 2013 at 14:58
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4$\begingroup$ Clever guys the Russians. The mistake went through in the Spanish translation :) $\endgroup$ Sep 24, 2013 at 15:05
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7$\begingroup$ @Adrián Barquero No, not all Russians are clever. :-) The book was translated by Yu.Manin. $\endgroup$ Sep 24, 2013 at 15:11
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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.