---- > [!definition] Definition. ([[ring adjunction]]) > Let $R \subset A$ be an inclusion of [[commutative ring|commutative]] [[ring|rings]], so $A$ is an $R$-[[algebra]]. The $R$-[[subalgebra]] [[subalgebra generated by a subset|generated by]] $x \in A$ is given by all combinations of powers of $x$ with coefficients in $R$, and is accordingly denoted $R[x]$. Because $R[x]$ is the smallest [[subring]] of $A$ containing both $R$ and $x$, we say that we have **adjoined $x$ to $R$**. > Feels a bit like [[constructing field extensions in which a given polynomial acquires a root]]. No time right now to try making that precise. ---- #### Let $R$ be a [[commutative ring|commutative]] [[ring]] and $S \subset R$ a [[subring]]. We may **adjoin $\alpha \in R$ to $S$** by considering the smallest [[subring]] of $R$ containing both $S$ and $\alpha$. It is not hard to see that such a ring is precisely given by all combinations of powers of $\alpha$ with coefficients in $S$, and it is accordingly denoted by $R[\alpha]$ as the image of $\alpha$ under the evaluation morphism $R[T] \to R$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```