I would like to know if there are results about the dimension of the Algebra generated by two commuting Matrices over a ring (as there are in the case of a Field). The good news is that "my" ring is commutative, the bad news is that it is not a domain of integrity. Thanks for your help
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$\begingroup$ If your ring is reduced, it embeds in a product of domains. In any case, it would be interesting to inspect the ring of quotients of your ring. $\endgroup$ – Luc Guyot Apr 13 at 15:12

4$\begingroup$ You should be more precise in what you want to call "dimension". $\endgroup$ – YCor Apr 13 at 19:15

$\begingroup$ More precise:I know the case of a field and results by Gerstenhaber, etc..., who proves that the Algebra generated by two commuting matrices (nxn rows and columns) is not greater than n; is it true when the matrices of their coefficients in a ring (not embeddeable (?) in a field ? $\endgroup$ – teller Apr 14 at 17:22