Abstract
The purpose of this note is to focus attention on a collection of algebras (over an arbitrary commutative ring k) for which we believe reasonable structure theorems should be obtainable. We call these algebras triangular. (§1). In §3 we show that they share a striking feature: their global dimension is finite. The definition of triangularity involves a new radical — which we call the tensor radical — for Noetherian rings. (§1). If A is commutative its tensor radical is 0 while for finite dimensonal algebras over a field the tensor radical is contained in the Jacobson radical. When k is a field a triangular k-algebra is a finite dimensional algebra satisfying: 1. the Jacobson radical, J(A), is a tensor nilpotent A-bimodule, i.e. a tensor product (over A) of sufficiently many copies of J(A) vanishes; 2. A/J(A) is a separable k-algebra. For example the algebra of upper triangular n × n matrices is triangular while k[x]/x2 is not.
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© 1988 Kluwer Academic Publishers
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Gerstenhaber, M., Schack, S.D. (1988). Triangular Algebras. In: Hazewinkel, M., Gerstenhaber, M. (eds) Deformation Theory of Algebras and Structures and Applications. NATO ASI Series, vol 247. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3057-5_7
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