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integral closure (Definition)

Let $ B$ be a ring with a subring $ A$. The integral closure of $ A$ in $ B$ is the set $ A' \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'$ of $ \mathbb{Z}$ is often called the ring of integers in $ B$.



"integral closure" is owned by djao.
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See Also: integrally closed

Also defines:  ring of integers

Attachments:
examples of ring of integers of a number field (Example) by alozano
the ring of integers of a number field is finitely generated over $\mathbb{Z}$ (Theorem) by alozano
ring of $S$-integers (Definition) by alozano
unique factorization and ideals in ring of integers (Theorem) by pahio
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Cross-references: integral, subring, ring
There are 81 references to this entry.

This is version 4 of integral closure, born on 2002-01-05, modified 2002-06-09.
Object id is 1299, canonical name is IntegralClosure.
Accessed 9687 times total.

Classification:
AMS MSC: 13B22 (Commutative rings and algebras :: Ring extensions and related topics :: Integral closure of rings and ideals ; integrally closed rings, related rings )
Pending Errata and Addenda
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