Data structures – Objects in different representations.

The succinct representation of a combinatorial object has been studied in the past. I am interested in the succinct representation of algebraic objects such as rings, fields, etc. I am new to the domain of thesis (succinct representation), so I am not very aware of the research questions related to the succinct representation of objects. For me, it seems that a succinct representation means a shorter or more compact representation of the objects with a slight or no increase in the execution time to perform the operations on an object.

Question: Has the succinct representation of algebraic objects been studied in the past? What could be the motivation behind the succinct representation of algebraic structures?