Ring Theory/Ring extensions

Definition (ring extension):

Whenever $S$ is a ring and $R\subset S$ is a subring of $S$, we say that $S$ is a ring extension of $R$ and write $S/R$.

Note that if $S/R$ is a ring extension, then $S[x]/R[x]$ is a ring extension; indeed, the set $S[x]$ is the set of all polynomials with coefficients in $S$, the set $R[x]$ is the set of all polynomials with coefficients in $R$, and $R[x]$ is a subring of $S[x]$.

Proposition (existence of splitting ring):

Let $R$ be a ring, and let $p\in R[x]$ be a polynomial over $R$. Then there exists a ring extension $S/R$ such that in ${\overline {S}}$, $p$ decomposes into linear factors, that is,

$p(x)=(x-\lambda _{1})\cdots (x-\lambda _{n})$ for certain $\lambda _{1},\ldots ,\lambda _{n}\in {\overline {S}}$.

Proof: We prove the theorem by induction on the degree of $p$. Suppose first that $p$ can be decomposed into two polynomials $\Box$