Journal of the
Korean Mathematical Society JKMS

Let $(B,m_{B},k)$ be a maximal commutative $k$-subalgebra of $M_{m}(k)$. Then, for some element $z \in Soc(B)$, a $k$-algebra $R=B[X,Y]/I$, where $I=(m_{B}X,m_{B}Y,X^{2}-z,Y^{2}-z,XY)$ will create an interesting maximal commutative $k$-subalgebra of a matrix algebra which is neither a $C_{1}$-construction nor a $C_{2}$-construction. This construction will also be useful to embed a maximal commutative $k$-subalgebra of matrix algebra to a maximal commutative $k$-subalgebra of a larger size matrix algebra.

Keywords: $C_{2}^{2}$-construction

MSC numbers: 15A27, 15A33