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I am new to semigroup research, so I apologize if this is an easy question.

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Assuming by M(n, Z) you mean the semigroup (monoid) of n × n matrices over the integers under multiplication: no, it is not even finitely generated, because the determinant M(n, Z) → Z is multiplicative (Z denoting the monoid of integers under multiplication) and Z is not finitely generated (by the infinitude of primes).

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    $\begingroup$ If n=0, then M(n,Z) is finitely presented! (Sorry for being obnoxious.) $\endgroup$
    – Bjorn Poonen
    Commented Feb 10, 2010 at 8:33

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