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Quantaloids and non-commutative ring representations. (English) Zbl 0847.18005
In an earlier paper, the author, together with F. Borceux, developed a generic sheaf representation for commutative rings based on the quantale of ideals of the ring (thus generalizing representations over frames) [see “A generic sheaf representation for rings”, in: Category theory, Lect. Notes Math. 1488, 30-42 (1991; Zbl 0743.18002)]. This was later generalized to include non-commutative rings by considering the quantale of two-sided ideals of the ring, however this quantale may not behave well for certain rings. The article under review proceeds by generalizing to the level of quantaloids, a natural categorical generalization of quantales, and to one particular example, which allows the author to simultaneously consider both left and right ideals of the ring, as well as the two-sided ones. A suitable notion of sheaf is obtained by considering idempotent matrices valued in this quantaloid, with morphisms certain adjoint pairs of matrices. [For a detailed look at quantaloids, matrices valued in them etc., see the reviewer’s book, “The theory of quantaloids”, Res. Notes Math. 348 (1996; Zbl 0845.18003).]
A sheaf representation is developed over the quantaloid of left and right ideals of the ring, and it is shown how both the center of the ring, as well as all the elements, can be suitably recovered from this sheaf. This paper represents an important step in investigating sheaf-like structures over quantaloids.

MSC:
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
16G99 Representation theory of associative rings and algebras
06F05 Ordered semigroups and monoids
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