Let \(Q\) be a finitely generated abelian group and \(R\) a commutative ring. The author constructs a primitively generated cocommutative differential graded Hopf algebra \(A(Q)\) whose homology algebra is isomorphic, as \(R\)-algebra, to the cohomology ring of \(Q\). He then uses this to obtain an explicit description of the integral cohomology ring of \(Q\). The construction of \(A(Q)\) is based on ideas which go back to work of Cartan, Eilenberg and Maclane [see e.g. \textit{S. Eilenberg} and \textit{S. Maclane}, Ann. Math. (2) 58, 55--106 (1953; Zbl 0050.39304) and ibid. 60, 49--139 (1954; Zbl 0055.41704)]. \(A(Q)\) is defined via a presentation of \(Q\) and is natural in the presentation only and not in the group itself. However it is quite a ``small'' model which is important for explicit computations.