# ascending chain condition

• Nov 11th 2009, 02:02 AM
knguyen2005
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
• Nov 11th 2009, 03:22 AM
tonio
Quote:

Originally Posted by knguyen2005
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 $\displaystyle R^{*}\setminus U(R)$ , anyway? For me, $\displaystyle R^{*}$ usually denotes the units of R...but also $\displaystyle U(R)$!

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

About $\displaystyle (a_1+1)\cdot...\cdot (a_t+1)$ : 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
• Nov 16th 2009, 02:01 PM
knguyen2005
Quote:

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

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

About $\displaystyle (a_1+1)\cdot...\cdot (a_t+1)$ : 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