Central multiples of a module are submodules

Let R be a ring with 1, let M be a left R-module, and let z be an element in the center of R. Prove that z \cdot M is a submodule of M. If F is a field, R = \mathsf{Mat}_2(F), M = R a left R-module under multiplication, and e = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}, then eR is not a left R-submodule of R.


Let z \in Z(R). We use the submodule criterion to show that z \cdot M is a submodule of M. Note first that z \cdot 0 \in z \cdot M, so that z \cdot M is not empty. Now suppose z \cdot x, z \cdot y \in z \cdot M and r \in R. Then (z \cdot x) + r \cdot (z \cdot y) = z \cdot x + rz \cdot y = z \cdot x + zr \cdot y = z \cdot (x + r \cdot y) \in z \cdot M. Thus z \cdot M is a submodule of M.

Now for the example, note that eR = \left\{ \begin{bmatrix} a & b \\ 0 & 0 \end{bmatrix} \ |\ a,b \in F \right\}. Letting a = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} and r = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, we have r \cdot (ea) = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix} \notin eR. Thus eR is not a submodule of R.

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