The Brauer Group of Irreducible Coalgebras

In this paper we solve a previously formulated conjecture (B. Torrecillas, F. Van Oystaeyen, and Y. H. Zhang, J. Algebra177 (1995), 568). That is, for a cocommutative irreducible coalgebra C, the homomorphism (−)*: Br(C) → Br(C*) is injective. The proof uses Morita–Takeuchi theory and the linear topology of all closed cofinite left ideals in C*. As an inmediate consequence, Br(C) is a torsion group. Some cases where the map (−)* is an isomorphism are studied. It is also deduded from the main result that the inclusion of the coradical C0 into C induces a monomorphism i*: Br(C) → Br(C0). New examples of Brauer groups of cocommutative coalgebras may be given using this fact.