# integral closure is ring

Theorem. Let $A$ be a subring of a commutative ring $B$ having nonzero unity. Then the integral closure^{} of
$A$ in $B$ is a subring of $B$ containing $A$.

*Proof.* Let $x$ be an arbitrary element of the integral closure ${A}^{\prime}$ of $A$ in $B$. Then there are the elements ${a}_{0},{a}_{1},\mathrm{\dots},{a}_{n-1}$ of $A$ such that

$${a}_{0}+{a}_{1}x+\mathrm{\dots}+{a}_{n-1}{x}^{n-1}+{x}^{n}=\mathrm{\hspace{0.33em}0}$$ |

where $n>0$. If $f(X)={c}_{0}+{c}_{1}X+\mathrm{\dots}+{c}_{m}{X}^{m}$ is a polynomial^{} in $A[X]$ with degree $m>n$, we have

$f(x)$ | $\mathrm{\hspace{0.33em}}={c}_{0}+{c}_{1}x+\mathrm{\dots}+{c}_{m-1}{x}^{m-1}+{c}_{m}{x}^{m-n}(-{a}_{0}-{a}_{1}x-\mathrm{\dots}-{a}_{n-1}{x}^{n-1})$ | ||

$\mathrm{\hspace{0.33em}}={c}_{0}^{\prime}+{c}_{1}^{\prime}x+\mathrm{\dots}+{c}_{m-1}^{\prime}{x}^{m-1}$ |

where the elements ${c}_{i}^{\prime}$ belong to $A$. This procedure may be repeated until we see that $f(x)$ is an element of the $A$-module generated by $1,x,\mathrm{\dots},{x}^{n}$. Accordingly,

$$A[x]=A+Ax+\mathrm{\dots}+A{x}^{n}$$ |

is a finitely generated^{} $A$-module.

Now we have evidently $A\subseteq {A}^{\prime}$. Let $y$ be another element of ${A}^{\prime}$. Then

$$A[x,y]=A[x][y]$$ |

is a finitely generated $A[x]$-module, whence $A[x,y]$ is a finitely generated $A$-module. Because the elements $x-y$ and $xy$ belong to $A[x,y]$, they are integral over $A$ and thus belong to ${A}^{\prime}$. Consequently, ${A}^{\prime}$ is a subring of $B$ (see the http://planetmath.org/node/2738subring condition).

## References

- 1 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press, New York (1971).

Title | integral closure is ring |
---|---|

Canonical name | IntegralClosureIsRing |

Date of creation | 2013-03-22 19:15:40 |

Last modified on | 2013-03-22 19:15:40 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13B22 |

Related topic | PolynomialRing |

Related topic | RingAdjunction |

Related topic | IntegralClosuresInSeparableExtensionsAreFinitelyGenerated |