> [!definition] > > Let $R$ be a [[Ring|ring]] and $E$, $F$ be $R$-[[Module|modules]]. Let $T^k(E)$ be the $k$-fold [[Tensor Product|tensor product]] of $E$, and $N \subset T^k(E)$ be a submodule generated by elements of the form > $ > \bigotimes_{j = 1}^k x_j \quad \exists (i, j): i \ne j, x_i = x_j > $ > Define $\bigwedge^k(E) = T^k(E)/N$ as the quotient module, then there exists a canonical [[Alternating Multilinear Map|alternating multilinear map]] > $ > \begin{CD} > E^k @>{\varphi}>> T^k(E) @>{\pi}>> T^r(E)/N = \bigwedge^k (E) > \end{CD} > $ > which is alternating. Moreover, it is [[Universal Object|universal]] with respect to the $k$-multilinear alternating maps on $E$. The canonical map combined with the quotient is known as the **alternating product**.