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In my life I saw lots of ways of constructing rings: polynomial rings, quotient rings, localizations, endomorphism rings, rings of fractions, integral closure of a ring, center of a ring, etc... These constructions are very useful to build up exotic counterexamples (for example rings being left noetherian and not right noetherian).

In these days I am studying some theory of valuation rings, and my professor showed me a new way of constructing rings.

Suppose $R \subseteq S$ is an extension of commutative rings (with unity), and $I$ is an ideal of $S$. Then the following is a subring of $S$: $$R+I = \{ r+i : r \in R, i \in I\}$$

This construction is used for example to build the ring $\Bbb{Z}+x\Bbb{Q}[[x]]$, which is a local domain of dimension $2$ dominated by $\Bbb{Q}[[x]]$.

I found this simple idea very easy to understand and actually it is useful to build up new things. However, I wonder why I never saw it in my life (for example it does not appear in Atiyah-MacDonald, or other books of commutative algebra).

My question is:

  • Does this construction have a name? Where can I find some references to study such rings?

  • Is there any reason why it is so poorly considered?

Crostul
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  • There isn't much 'construction' to it: it's just a subring of a ring you already have in hand. That being the case, it's very frequently used in building examples, too common to have a name. from the previous comment, it sounds like this particular choice of generators nevertheless has some interesting properties beyond your run of the mill subring. – rschwieb May 31 '15 at 11:33
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    There is a closely connected construction that does have a name: If $K$ is a commutative ring with unity, and if $S$ is a nonunital $K$-algebra, then $K \oplus S$ becomes a (unital!) ring when we define multiplication by $\left(k, s\right) \left(k', s'\right) = \left(kk', ks' + k's + ss'\right)$. Its unity is $\left(1, 0\right)$, and the map $K \to K \oplus S, \ k \mapsto \left(k, 0\right)$ is a (unital) ring homomorphism, making it into a $K$-algebra. This is called a Dorroh extension. Your construction is a variant where the sum is not direct and $S$ is already an ideal of a ring. – darij grinberg Jun 01 '15 at 15:28
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    Some cases are heavily studied, e.g. see the links I gave on the $D+M$ construction (esp. Zafrullah's survey) in this answer. – Bill Dubuque Jul 22 '17 at 22:00

1 Answers1

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If $A$ is an algebraic structure, $S\leq A$ is a subalgebra, and $\theta$ is a congruence on $A$, then the saturation of $S$ with respect to $\theta$ is the set

$$S^{\theta} = \{a\in A\;|\;\exists s\in S((a,s)\in\theta)\}.$$

$S^{\theta}$ is the smallest subalgebra of $A$ containing $S$ that is a union of $\theta$-classes. This construction comes up early in one's mathematical education because of its role in the second isomorphism theorem, which is the assertion that $S/(\theta|_S)\cong S^{\theta}/(\theta|_{S^{\theta}}).$ (Here $\theta|_S$ denotes the restriction of $\theta$ to $S$.)

To answer your first question, $R+I$ is the saturation of $R$ with respect to (the congruence associated to) $I$. You can learn about the saturation of a subalgebra by a congruence by googling "saturation" of a subalgebra by a congruence.

Keith Kearnes
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