Abstract
Our aim is to generalize to the non-commutative case, the generic representation of commutative rings by sheaves on their quantales of ideals. As the quantale of two-sided ideals is not a sufficiently rich structure, we define and work on a quantaloid of left and right ideals. A workable notion of sheaf is introduced using matrices with values in a quantaloid. For a given ringR, we obtain a category of sheaves where the terminal object is endowed with a special subobject. There exists a representing sheaf forR in the sense that the elements ofR correspond to the sections from the special subobject and the global sections correspond to the center.
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van den Bossche, G. Quantaloids and non-commutative ring representations. Appl Categor Struct 3, 305–320 (1995). https://doi.org/10.1007/BF00872902
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DOI: https://doi.org/10.1007/BF00872902