Let be a discrete group, a commutative unital ring, the group ring of and a left -module. Then there is well-defined -module called the module of -coinvariants. Here denotes the smallest sub--module of containing all expressions of the form where and .
Let be a -coalgebra, a group-like element, that is, an element such that , and a right -coaction. Any element such that is called a -coinvariant element in the -comodule . Suppose is a bialgebra, an algebra and a coaction making into a right -comodule algebra. The unit element is a group-like element, and we call -coinvariants simply -coinvariants. The subset of -coinvariants in is a subalgebra, called the subalgebra of coinvariants.