Anarchy literally means "No Rulers". It does NOT mean "No Rules". This has many ramifications.
It can mean rules, so long as they're not enforced onto an unwilling population by a ruling minority. In essence, enforcement of a rule is primarily in the interest of a community managing its own well-being, than it can be compatible with anarchy. If enforcement of a rule is primarily for the benefit of one section of the community at the cost of another, it most likely represents a form of rulership, and is incompatible with anarchy.
It can mean government, so long as that government is strictly organizational, and not authoritative. When the government enforces rules that are negotiated and agreed to by all effected parties, it is compatible with anarchy. When the government creates rules and then enforces them, and those effected by them have little to no say, it is incompatible with anarchy. It is rulership.
Yes, the above leaves plenty of room for grey areas. Such is life. The very nature of an ideal is that it isn't obtainable. Everyone has their ideologies, and all ideologies are based on ideals, hence the very root of the word. The most common response to a post like this is "Such a utopian ideal is so naive!", but I could say that about literally every political stance ever.
The very concept of a person being rich is incompatible with anarchy. Wealth is a form of authority that is unavoidably hierarchical and generally not mutually agreed upon by the whole community that is effected by this distribution. So no. If there are rich people to bribe and force their way into a position of rulership, then there was no anarchy in the first place.