Ring extension where primes can be lifted is integral?
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Let $B/A$ be a ring extension of unital commutative rings.
Suppose for each prime $psubset A$, there is $q subset B$ prime with $q cap A = p$.
It is not true that $B$ is integral over $A$, for instance if we take a transcedental extension of a field, then it's just that there's no nonzero prime ideals.
My question is how close is the relationship between integrability and lifting primes, meaning what conditions can we add, so that a ring extension satisfying it is an integral extension?
commutative-algebra
asked Nov 16 at 10:41
Andy
485513
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up vote
0
down vote
favorite
Let $B/A$ be a ring extension of unital commutative rings.
Suppose for each prime $psubset A$, there is $q subset B$ prime with $q cap A = p$.
It is not true that $B$ is integral over $A$, for instance if we take a transcedental extension of a field, then it's just that there's no nonzero prime ideals.
My question is how close is the relationship between integrability and lifting primes, meaning what conditions can we add, so that a ring extension satisfying it is an integral extension?
commutative-algebra
asked Nov 16 at 10:41
Andy
485513
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $B/A$ be a ring extension of unital commutative rings.
Suppose for each prime $psubset A$, there is $q subset B$ prime with $q cap A = p$.
It is not true that $B$ is integral over $A$, for instance if we take a transcedental extension of a field, then it's just that there's no nonzero prime ideals.
My question is how close is the relationship between integrability and lifting primes, meaning what conditions can we add, so that a ring extension satisfying it is an integral extension?
commutative-algebra
asked Nov 16 at 10:41
Andy
485513
Let $B/A$ be a ring extension of unital commutative rings.
Suppose for each prime $psubset A$, there is $q subset B$ prime with $q cap A = p$.
It is not true that $B$ is integral over $A$, for instance if we take a transcedental extension of a field, then it's just that there's no nonzero prime ideals.
My question is how close is the relationship between integrability and lifting primes, meaning what conditions can we add, so that a ring extension satisfying it is an integral extension?
commutative-algebra
commutative-algebra
asked Nov 16 at 10:41
Andy
485513
asked Nov 16 at 10:41
Andy
485513
asked Nov 16 at 10:41
Andy
485513
asked Nov 16 at 10:41
Andy
485513
asked Nov 16 at 10:41
Andy
485513
485513
add a comment |
add a comment |
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