Consider Marshall Stone's representation theorem:
I would like to know in which specific way, if any, it is connected with the determination of right or left one-sided ideals in a ring. Simple examples are welcome.
2026-04-15 10:09:18.1776247758
Stone representation theorem and right(or left-) one-sided ideals in a ring
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It can tell you a bit about the ideals of the Boolean algebra you are representing, yes. When the Boolean algebra is representable by finitely many copies of $F_2$, then you can recover all ideals: every ideal of $\prod _{i=1}^n F_2$ is of the form $\prod _{i=1}^n I_i$ where each $I_i$ is either $\{0\}$ or $F_2$.
When there are infinitely many copies of $F_2$, it is no longer so easy to describe the ideals.
Strictly one-sided ideals are irrelevant in Boolean algebras since Boolean algebras are all commutative, and all ideals are two-sided.