Proposition 1

Let $M_1$ be an $A_1$-module and $M_2$ an $A_2$-module, neither reduced to 0. If the module $M = M_1\otimes M_2$ over the ring $A = A_1\otimes A_2$ is simple (resp. isotypical, semisimple), then the $A_1$-module $M_1$ and the $A_2$-module $M_2$ are simple (resp. isotypical, semisimple).

Suppose that M is a semisimple A-module. Let $N_1$ be an $A_1$-submodule of $M_1$. Denote the canonical image of $N_1\otimes M_2$ in the A-module M by N. By Corollary 2 of VIII, p. 56, there exists an A-linear projector $p$ in M with image N. By assumption, we have $M_2\not= 0$; we can therefore choose an element $m$ of $M_2$ and a linear form $\varphi$ on the K-vector space $M_2$ such that $\varphi (m) = 1$ (II, §7, No. 5, p. 302, Corollary 2). Let $u$ be the mapping from $M_1$ to M defined by $u(m_1) =m_1\otimes m$, and let $v$ be the K-linear mapping from M to $M_1$ characterized by $v(m_1\otimes m_2) =\varphi (m_2)m_1$. Set $q=v\circ p\circ u$. The mapping $q: M_1\rightarrow M_1$ is $A_1$-linear, its image is contained in $N_1$, and we have $q(n) =n$ for every $n\in N_1$. Consequently, $q$ is a projector in $M_1$ with image $N_1$. We have proved that $M_1$ is a semisimple $A_1$-module (VIII, p. 56, Corollary 2).

Suppose that M is simple and that $M_1$ is the direct sum of two $A_1$-submodules $M'_1$ and $M''_1$. Set $M'= M'_1\otimes M_2$ and $M''= M''_1\otimes M_2$; then M is the direct sum of the A-submodules $M'$ and $M''$. Since M is simple, $M'$ or $M''$ must be reduced to 0; as we have $M_2\not= 0$ by assumption, we have $M'_1= 0$ or $M''_1= 0$ (II, §3, No. 7, p. 256, Corollary 2). This proves that $M_1$ is a simple $A_1$-module.

Now suppose that M is an isotypical A-module. Let S and T be simple $A_1$-submodules of $M_1$. The A-modules $S\otimes M_2$ and $T\otimes M_2$ can then be identified with nonzero submodules of M. They are therefore isotypical of the same type as M. By Remark VIII, p. 61, there exists a nonzero A-linear mapping $f: S\otimes M_2\rightarrow T\otimes M_2$. The mapping $f$ is, in particular, $A_1$-linear. Since the $A_1$-modules $S\otimes M_2$ and $T\otimes M_2$ are nonzero and isotypical of type S and T, respectively, S and T are isomorphic. This proves that $M_1$ is an isotypical $A_1$-module.