Study of rings of the form $R+I$
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?
Solution 1:
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.