Integrally closed domain

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Many well-studied domains are integrally closed: Fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed.

To give a non-example, let k be a field and ${\displaystyle A=k[t^{2},t^{3}]\subset B=k[t]}$ (A is the subalgebra generated by t² and t³.) A and B have the same field of fractions, and B is the integral closure of A (since B is a UFD.) In other words, A is not integrally closed. This is related to the fact that the plane curve ${\displaystyle Y^{2}=X^{3}}$ has a singularity at the origin.

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