Tensor product of modules over non-unital algebras and Lie-Rinehart algebras

Jan Kubarski Institute of Mathematics, Technical University of Łódź

In 1990-99 J.Huebschmann wrote a series of papers relating to Lie-Rinehart algebras. In J. Huebschmann, Poisson cohomology and quantization, J. für die Reine und Angew. Math. 408 (1990), 57-113.the base of the series - the author wrote [p.70] ”Let A be an algebra over R, not necessarily with 1” and repeated this sentence in the context of commutative algebras on next pages. However, all the technical tools which were used, are appropriate in the case of unital algebras only. If we consider the typical situation where the base ring R is unital, then non-unitality of the R-algebra A means that there is no homomorphism of rings l : R → A such that l(r) · a = r · a = a · l(r). There are some simple anomalies in the theory of A-modules over non-unital R-algebra A which caused that the planned researches on Lie-Rinehart algebras for algebras not necessarily with 1 failed. Moreover, the construction of the Picard group for a ring A with a unit can not be adapted to the non-unital case. The reason is that there is a difference between projective modules in the category of non-unital and unital modules. The aim of this paper is to construct the notion of a tensor product of modules over non-unital algebras which does not possesses the anomalies and its applications for the Picard group of non-unital algebras and Lie-Rinehart algebras.