Definition of $\mathbf{Z}$-multilinear mapping

Question

Let $(G_{\lambda})_{\lambda\in L}$ be a (not necessarily finite) family of $\mathbf{Z}$-modules, $H$ a $\mathbf{Z}$-module and $u:\prod_{\lambda\in L}G_\lambda\rightarrow H$. If for each $\mu\in L$, $x,y\in G_\mu$ and $z\in\prod_{\lambda\ne\mu}G_{\lambda}$ $$u(x+y,(z_{\lambda})_{\lambda\ne\mu})=u(x,(z_{\lambda})_{\lambda\ne\mu})+u(y,(z_{\lambda})_{\lambda\ne\mu}),$$ where $\prod_{\lambda\in L}G_\lambda$ is identified with $G_\mu\times\prod_{\lambda\ne\mu}G_\lambda$, then $u$ is called $\mathbf{Z}$-multilinear.

Is there a way to transform this definition into one that doesn't identify $\prod_{\lambda\in L}G_\lambda$ with $G_\mu\times\prod_{\lambda\ne\mu}G_\lambda$?

Answer 1

This is effectively just a rephrasing of the definition you gave without using the identification between $\prod_{\lambda\in L}G_\lambda$ and $G_\mu\times\prod_{\lambda\neq\mu}G_\lambda$.

Let $(G_{\lambda})_{\lambda\in L}$ be a (not necessarily finite) family of $\mathbf{Z}$-modules, $H$ a $\mathbf{Z}$-module and $u:\prod_{\lambda\in L}G_\lambda\rightarrow H$. If for each $\mu\in L$ and $z\in\prod_{\lambda\in L}G_{\lambda}$, we have $$u((z_{\lambda})_{\lambda\in L})=u((z'_{\lambda})_{\lambda\in L})+u((z''_{\lambda})_{\lambda\in L}),$$ whenever $z_\mu=z'_\mu+z''_\mu$ and $z_\lambda=z'_\lambda=z''_\lambda$ for all $\lambda\neq \mu$, then $u$ is called $\mathbf{Z}$-multilinear.