Abstract
It is shown that the ring of two 2×2 generic matrices over a field has infinite global dimension. It is also proved that there is a non-free projective module over that ring. Finally, the authors show that the trace ring of that generic matrix ring is an iterated Ore extension from which it follows that the trace ring has global dimension five and that the finitely-generated projective modules are stably free.
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Small, L.W., Stafford, J.T. Homological properties of generic matrix rings. Israel J. Math. 51, 27–32 (1985). https://doi.org/10.1007/BF02772956
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DOI: https://doi.org/10.1007/BF02772956