So I am thinking about the structure of a commutative ring $R$. Give a family of principal ideals $A=\{(x_{i})\}$, and if $R=\bigcup_{i}(x_{i})$, is it true that there are finite number of principal ideals in $A$ such that $(1)=(x_{1},x_{2},...,x_{n})$? Actually, this comes from the statement that "Affine schemes are quasi-compact". Can anyone help? Thanks!
2026-03-30 08:14:51.1774858491
Is it true that the generators of a commutative ring is finite?
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If $R = \bigcup_i (x_i)$, then $1\in (x_i)$ for some $i$, so one of the principal ideals must be $(1)=R$.