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One must verify this to ensure the scalar multiplication is well-defined. Since $xy+yx$ is in the ideal $I$, its coset is the zero element in $R$.

Conditions: $R = \mathbb{C}\langle x,y\rangle / \langle xy+yx\rangle$.; The proposed action is $(x+I)v=Xv$ and $(y+I)v=Yv$.; $X,Y$ satisfy $XY+YX=0$.

One must verify this to ensure the scalar multiplication is well-defined. Since $xy+yx$ is in the ideal $I$, its coset is the zero element in $R$.

Conditions: $R = \mathbb{C}\langle x,y\rangle / \langle xy+yx\rangle$.; The proposed action is $(x+I)v=Xv$ and $(y+I)v=Yv$.; $X,Y$ satisfy $XY+YX=0$.

The coset of $x$ acts as left multiplication by the matrix $X$, and the coset of $y$ acts as left multiplication by the matrix $Y$. This defines the scalar multiplication for the $R$-module structure on $\mathbb{C}^n$.

Conditions: $X, Y \in M_{n\times n}(\mathbb{C})$ satisfy $XY+YX=0$.; $R = \mathbb{C}\langle x,y\rangle / \langle xy+yx\rangle$.; $v \in \mathbb{C}^n$.

The coset of $x$ acts as left multiplication by the matrix $X$, and the coset of $y$ acts as left multiplication by the matrix $Y$. This defines the scalar multiplication for the $R$-module structure on $\mathbb{C}^n$.

Conditions: $X, Y \in M_{n\times n}(\mathbb{C})$ satisfy $XY+YX=0$.; $R = \mathbb{C}\langle x,y\rangle / \langle xy+yx\rangle$.; $v \in \mathbb{C}^n$.