A question about the integral extension of a integral domain.
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Suppose that $R$ is an integral domain, $S$ is integral over $R$, that is to say, any elements in $S$ satisfies a monic polynomial with coefficients in $R$. (Here $S$ is a commutative ring with identity.) Is it right that $S$ is also an integral domain, or there is any counterexamples?
commutative-algebra integral-domain integral-extensions
asked Nov 27 at 3:12
Hugo
1559
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up vote
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Suppose that $R$ is an integral domain, $S$ is integral over $R$, that is to say, any elements in $S$ satisfies a monic polynomial with coefficients in $R$. (Here $S$ is a commutative ring with identity.) Is it right that $S$ is also an integral domain, or there is any counterexamples?
commutative-algebra integral-domain integral-extensions
asked Nov 27 at 3:12
Hugo
1559
1
Take $S=R[x]/(x^2)$.
– Mohan
Nov 27 at 3:30
add a comment |
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Suppose that $R$ is an integral domain, $S$ is integral over $R$, that is to say, any elements in $S$ satisfies a monic polynomial with coefficients in $R$. (Here $S$ is a commutative ring with identity.) Is it right that $S$ is also an integral domain, or there is any counterexamples?
commutative-algebra integral-domain integral-extensions
asked Nov 27 at 3:12
Hugo
1559
Suppose that $R$ is an integral domain, $S$ is integral over $R$, that is to say, any elements in $S$ satisfies a monic polynomial with coefficients in $R$. (Here $S$ is a commutative ring with identity.) Is it right that $S$ is also an integral domain, or there is any counterexamples?
commutative-algebra integral-domain integral-extensions
commutative-algebra integral-domain integral-extensions
asked Nov 27 at 3:12
Hugo
1559
asked Nov 27 at 3:12
Hugo
1559
asked Nov 27 at 3:12
Hugo
1559
asked Nov 27 at 3:12
Hugo
1559
asked Nov 27 at 3:12
Hugo
1559
1559
1
Take $S=R[x]/(x^2)$.
– Mohan
Nov 27 at 3:30
add a comment |
1
Take $S=R[x]/(x^2)$.
– Mohan
Nov 27 at 3:30
1
1
Take $S=R[x]/(x^2)$.
– Mohan
Nov 27 at 3:30
Take $S=R[x]/(x^2)$.
– Mohan
Nov 27 at 3:30
add a comment |
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Take $S=R[x]/(x^2)$.
– Mohan
Nov 27 at 3:30