# Ring Theory/Ring extensions

Definition (ring extension):

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

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

Proposition (existence of splitting ring):

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

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

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