I am looking at this problem at the moment: If R is a commutative ring with 1, in which every ideal is the sum of finitely many principal ideals $(I=r_1R+r_2R+...+r_nR)$ , show that this implies, that any ascending chain of Ideals $(I_1 ⊂ I_2 ⊂ I_3 ⊂ ...)$ is finite. Is the opposite implication also true? I honestly am not sure how to show either part of the question. Does anyone have any ideas?
2026-03-28 05:23:16.1774675396
Every ideal is the sum of principal ideals implies ascending chain of ideals is finite?
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A ring a is called Noetherian if any ascending chain of ideal terminates. This property is equivalent to saying that every ideal is finitely generated. So if an ideal is finitely generated, then it is the sum of finitely many principal ideals. This is because if $I=\langle a_1,\dots,a_n\rangle$, then $I=\langle a_1,\dots,a_n\rangle = \langle a_1\rangle+\langle a_2\rangle+\dots+\langle a_n\rangle$ = sum of finitely many principal ideals.
It is also easy to see that the converse is also true. If every ideal is a sum of finitely many principal ideals,then it is finitely generated. Hence, the ring is Noetherian, meaning that every ascending chain of ideals terminates.