A cohomology ring is a mathematical structure used in algebraic topology to study the properties of topological spaces. It combines the concepts of cohomology, which assigns algebraic invariants to spaces, and ring theory, where elements can be added and multiplied. The cohomology ring captures information about the space's shape and structure through its cohomology classes, which can be thought of as generalized "functions" on the space. In a cohomology ring, the elements correspond to cohomology classes, and the ring operations reflect the way these classes interact. The product of two classes represents a new class, while the addition combines classes. This structure helps mathematicians understand complex spaces by revealing relationships between their topological features, such as those studied by Henri Poincaré and Alexander Grothendieck.