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







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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







    share|cite|improve this question
























      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







      share|cite|improve this question













      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






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 16 at 10:41









      Andy

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