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It is known that a ring is left Noetherian if and only if every left -module has an injective (pre)cover. We show that if is a right -coherent ring, then every right -module has an -injective (pre)cover; if is a ring such that every -injective right -module is -pure extending, and if every right -module has an -injective cover, then is right -coherent. As applications of these results, we give some characterizations of -rings, von Neumann regular rings and semisimple rings....
The main result of the note is a characterization of 1-amenability of Banach algebras of approximable operators for a class of Banach spaces with 1-unconditional bases in terms of a new basis property. It is also shown that amenability and symmetric amenability are equivalent concepts for Banach algebras of approximable operators, and that a type of Banach space that was long suspected to lack property 𝔸 has in fact the property. Some further ideas on the problem of whether or not amenability (in...
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