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# integral closure

Let $B$ be a ring with a subring $A$. The integral closure of $A$ in $B$ is the set $A^{{\prime}}\subset B$ consisting of all elements of $B$ which are integral over $A$.

It is a theorem that the integral closure of $A$ in $B$ is itself a ring. In the special case where $A=\mathbb{Z}$, the integral closure $A^{{\prime}}$ of $\mathbb{Z}$ is often called the ring of integers in $B$.

Defines:

ring of integers

Related:

IntegrallyClosed

Type of Math Object:

Definition

Major Section:

Reference

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## Mathematics Subject Classification

13B22*no label found*

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## Attached Articles

examples of ring of integers of a number field by alozano

the ring of integers of a number field is finitely generated over $\mathbb{Z}$ by alozano

ring of $S$-integers by alozano

unique factorization and ideals in ring of integers by pahio

congruence in algebraic number field by pahio

integral closure is ring by pahio

the ring of integers of a number field is finitely generated over $\mathbb{Z}$ by alozano

ring of $S$-integers by alozano

unique factorization and ideals in ring of integers by pahio

congruence in algebraic number field by pahio

integral closure is ring by pahio