The modular group algebra of an elementary abelian p-group is isomorphic to the restricted enveloping algebra of a commuta-tive restricted Lie algebra. The different ways of regarding this algebra result in different Hopf algebra structures that determine cup products on cohomology of modules. However, it is proved in this paper that the products with elements of the polynomial subring of the cohomol-ogy ring generated by the Bocksteins of the degree one elements are independent of the choice of these coalgebra structures.