is a parent like a set?

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Yes! A parent indeed models a mathematical set, more often than not endowed with some structure (a ring, a poset, a vector space, ...).

In your case, you want to implement a class that inherits from Parent, and sets its category to Rings() during the initialization. We don't have yet a minimal example of implementation of a ring, but you can already look at:

sage: S = Monoids()
sage: M = S.example()
sage: M??