Abstract
This chapter discusses the algebras with Hochschild dimension ≤ 1. It is assumed that A is algebra over a commutative ring K. It is supposed that A0 is the opposite K-algebra to A, and A ® A0 is their tensor product over K, the enveloping algebra of A. A can be regarded as a left A ® A0-module in the natural way. The homological dimension of this module is called the Hochschild dimension of A and is denoted by K-dim A. K-dim A = 0 means that A is A ® A0-projective and A is separable over K. It is proved that the assumption that K is a Hensel ring makes it possible to replace the condition that h is a singular epimorphism by the much weaker condition that ker h is in the (Jacobson) radical of B, provided both B and C are assumed to be finitely generated as K-modules.
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