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Math Help - ascending chain condition

  1. #1
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    ascending chain condition

    Let R be a UFD, a in R*\U(R), a =p_1^(a_1).......p_t^(a_t) for distinct non-associated primes p_j, (a_j) >= 1, 1=<j=<t.

    If d|a the show that d~p_1^(b_1).....p_t^(b_t) for some 0=<b_j=<a_j, 1=<j=<t.

    Show that there are (1+a_1)....(1+a_t) distinct ideals (d) such that (a)=< (d). Deduce that R satisfies ascending chain condition (ACC) on principal ideals.

    I dont know how to do this question, please help

    Many thanks
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    Let R be a UFD, a in R*\U(R), a =p_1^(a_1).......p_t^(a_t) for distinct non-associated primes p_j, (a_j) >= 1, 1=<j=<t.

    If d|a the show that d~p_1^(b_1).....p_t^(b_t) for some 0=<b_j=<a_j, 1=<j=<t.

    Show that there are (1+a_1)....(1+a_t) distinct ideals (d) such that (a)=< (d). Deduce that R satisfies ascending chain condition (ACC) on principal ideals.

    I dont know how to do this question, please help

    Many thanks

    What is R^{*}\setminus U(R) , anyway? For me, R^{*} usually denotes the units of R...but also U(R)!

    Anyway: d\mid a \Longrightarrow a=\prod\limits_{i=1}^tp_i^{a_i}=xd\,,\,\,x\in R\Longrightarrow\,\,\,if\,\,q\mid d\,,\,\,q\,\,a\,\,prime\,\,then\,\,q\mid p_i for some  1\leq i\leq t. Continue from here.

    About (a_1+1)\cdot...\cdot (a_t+1) : think of natural numbers: if a were a natural number, then the above product would be the number of positive divisors of a...take it from here.

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    What is R^{*}\setminus U(R) , anyway? For me, R^{*} usually denotes the units of R...but also U(R)!

    Anyway: d\mid a \Longrightarrow a=\prod\limits_{i=1}^tp_i^{a_i}=xd\,,\,\,x\in R\Longrightarrow\,\,\,if\,\,q\mid d\,,\,\,q\,\,a\,\,prime\,\,then\,\,q\mid p_i for some  1\leq i\leq t. Continue from here.

    About (a_1+1)\cdot...\cdot (a_t+1) : think of natural numbers: if a were a natural number, then the above product would be the number of positive divisors of a...take it from here.

    Tonio
    Thank you very much Tonio
    I denote R*\U(R) is non-zero,non-unit

    I still can't figure out how to do the last part of the question. Please help
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