Abstract
It is well known that in the category RMod of left R-modules over a ring R with unit each X∈RMod is canonically a direct limit of copies of Rn=R⊺ ... ⊺R (n summands), where n is a fixed number ≥2. In other words, Rn is dense in RMod in the sense of Gabriel-Ulmer [4]. However, R=RI itself is in general not dense in RMod. The aim of this note is to give a partial answer to the question which rings R are dense in RMod and which are not. Typical examples of dense rings are (arbitrary) products of matrix rings, whereas any commutative ring ≠0 is not dense.
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Bäni, W. Ueber Ringe, Welche Dicht in Ihrer Modulkategorie Sind. Manuscripta Math 10, 379–394 (1973). https://doi.org/10.1007/BF01527260
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DOI: https://doi.org/10.1007/BF01527260