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If $K$ is a field extension of a field $F$ (that is, $F\subseteq K$), we write $K/F$. But if $R$ is a ring extension of the ring $S$ (that is, $S\subseteq R$), what is the equivalent notation? Do we write $R/S$ like we would for fields or is something like $R\supseteq S$ more appropriate?

Math Rules
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1 Answers1

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I'm not aware of any generally accepted definition of a "ring extension" with no further qualifiers. When you go beyond the case of fields it's much less clear that "an injection $S \to R$" is the correct notion of "ring extension." On the one hand you may want to discuss an arbitrary map $f : S \to R$, not necessarily injective; equivalently, you may want $R$ to be an arbitrary $S$-algebra. This is the most general thing you could ask for and has nice categorical properties. On the other hand you may want to require various properties of the map $f$: for example, that it's

  • integral (which generalizes algebraic extensions)
  • flat (automatic if $S$ is a field)
  • faithfully flat (various nice properties)
  • etale (also very nice properties but very restrictive)

and lots and lots of other possibilities. I am also not aware of notation for any of these. People just say "let $f : S \to R$ be (whatever)." (Also, it would be standard to write $f : R \to S$ here.)

Qiaochu Yuan
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  • I didn't think a ring extension was a mapping. If I understood correctly, $\mathbb{Q}(\zeta)$ was a field extension of $\mathbb{Q}$, while $\mathbb{Z}[\zeta]$ was a ring extension of $\mathbb{Z}$. Am I getting the terminology mixed up here? – Math Rules Nov 12 '20 at 01:02
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    @Math: I am just not aware of a generally accepted definition of "ring extension" one way or another. $\mathbb{Z}[\zeta]$ is an integral extension of $\mathbb{Z}$ and "integral extension" is a generally accepted term. – Qiaochu Yuan Nov 12 '20 at 02:21
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    If you search on "extension of rings" you will see that it is in wide use, e.g. "suppose $,R\subseteq S$ is an extension of rings..." – Bill Dubuque Nov 12 '20 at 02:43