The word for a state where the rich people are in charge is "plutocracy", from the Greek "πλοῦτος" meaning wealth. It directly translates to rule by the wealthy.
It is not "oligarchy". That comes from "ὀλίγος" meaning few, just as "monarchy" comes from "μόνος" meaning alone/sole/solitary. These words are principally about the number of rulers, not what kind of people they are.
Nor is it "aristocracy" from "ἄριστος" meaning "best"/"noblest", and is thus "the nobility". Usually self-styled, of course; and hence why "kleptocracy" from "κλέπτης", meaning thief, exists. Not everyone agrees with the narrative that bravery, moral superiority, and nobility are the foundations. (-:
Interestingly, "-cracy" from " -κρατία" (power) is a lot more productive as a suffix than "-archy" from "-αρχίᾱ" (rule). "panarchy" from "πᾶς" (all) has not really caught on.
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