Proposition 2

Let A be a ring, and let M be an A-module. Let I be a set. Then the bicommutant of the A-module $M^{(I)}$ coincides with the ring of homotheties of the $A''_M$-module $M^{(I)}$.

For $i\in I$, we denote the projection homomorphism from $M^{(I)}$ to M by $\pi_i: (x_j)_{j\in I}\mapsto x_i$ and the corresponding canonical injection (I, §4, No. 9, p. 47) by $\iota_i: M\rightarrow M^{(I)}$.

Let $u$ be an element of End$_A(M^{(I)})$. For all $i, j\in I$, the composition $u_{i,j}=\pi_j\circ u\circ \iota_i$ belongs to the commutant $A'_M$ of M. For every element $b$ of $A''_M$ and every $(x_i)\in M^{(I)}$, we have the relations

The homothety $b_{M^{(I)}}$ therefore belongs to the bicommutant of the A-module $M^{(I)}$.

Conversely, let $b$ be an element of the bicommutant of $M^{(I)}$. For $i, j\in I$, set $b_{i,j}=\pi_j\circ b\circ \iota_i$. Take $i, j\in I$ with $i\not=j$. Since $\iota_j\circ \pi_j$ belongs to the commutant of the A-module $M^{(I)}$, we have

Likewise, we have

Moreover, $b_{i,i}$ belongs to $A''_M$. It follows that $b$ coincides with a homothety of the $A''_M$-module $M^{(I)}$.