Group algebras whose simple modules are injective

Daniel R. Farkas, Robert L. Snider
1974 Transactions of the American Mathematical Society  
Let F be either a field of char 0 with all roots of unity or a field of char/i > 0. Let G be a countable group. Then all simple /{Gj-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The condition that all simple modules over a ring be injective first appeared in a theorem due to Kaplansky: a commutative ring satisfies the condition if and only if it is von Neumann regular. Several people have studied the
more » ... erty for noncommutative rings, a recent example being [3] . The authors of that paper suggest the problem of characterizing group algebras with this condition. In this paper we make substantial progress by offering Theorem 3. Let F be either a field of char 0 with all roots of unity or a field of char p > 0. Let G be a countable group. Then all simple F[G]-modules are injective if and only if G is locally finite with no elements of order char F and G has an abelian subgroup of finite index. The proof is divided into three parts. In §1, we show that F is injective as an F[G]-module if and only if G is locally finite with no elements of order char F. In the second and crucial section, we show that for a certain class of rings ("locally Wedderburn algebras") the condition that all simple modules are injective is equivalent to the property that all simple modules are finite dimensional over their commuting rings. In §3, we prove the main theorem by showing that if all simple modules are finite dimensional over their commuting rings then G is abelianby-finite. We would like to thank D.S. Passman for suggestions that shortened and improved our work.
doi:10.1090/s0002-9947-1974-0357475-5 fatcat:fdkb2dpw4jcqlnr2hiipe54moe