Let $R$ be a finite commutative ring with identity. Considere the matrix ring $A=M_n(R)$. What is the order of a maximal commutative subring of $A$ that contains all scalar matrices?

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`$\left\{\left(\begin{array}{cc}a&b\\ 0&a\end{array}\right)\mid a \in R, b \in I\right\}$`

is a subalgebra of $M_2(R)$ which is neither free nor finitely generated. What dimension would you assign to it? – Florian Eisele Aug 7 '12 at 13:35