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Group algebras in which complements are summands. (Russian, English) Zbl 1110.16024

Fundam. Prikl. Mat. 11, No. 3, 3-11 (2005); translation in J. Math. Sci., New York 144, No. 2, 3875-3880 (2007).
Let \(G\) be a group, and let \(KG\) be its group algebra over some field \(K\). A ring \(R\) is said to be almost self-injective if for any right ideal \(H\) of \(R\) and every \(R\)-homomorphism \(f\colon H\to R\), there exists an \(R\)-homomorphism \(g\colon R\to R\) such that \(g\circ f=\text{id}_H\). The paper studies the following problems: (1) When is \(KG\) almost self-injective? (2) When is \(KG\) continuous? The authors show that every almost self-injective algebra is self-injective, and if \(KG\) is continuous, then \(G\) is locally finite.

MSC:

16S34 Group rings
16D50 Injective modules, self-injective associative rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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