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. Algebra 177 (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 immediate conseque nce, Br(C) is a torsion group. Some cases where the map(-)* is an isomorphi sm are studied. It is also deduded from the main result that the inclusion of the coradical C-0 into C induces a monomorphism i*: Br(C)--> Br (C-0). N ew examples of Brauer groups of cocommutative coalgebras may be given using this fact. (C) 2001 Academic Press.