What is perfect rings?

Perfect rings are a type of mathematical object called algebraic systems. More specifically, they are commutative rings (meaning the order in which numbers are multiplied does not matter) in which every element can be expressed as the product of idempotents (elements that are equal to their own squares). This allows perfect rings to have a number of useful properties, such as having finite decomposition into direct summands and being isomorphic to its own opposite ring. They have applications in algebraic geometry, representation theory, and number theory, among other areas of mathematics. Some examples of perfect rings include fields, finite dimensional algebras over a field, and polynomial rings over a field with characteristic p raised to a power of a prime.