I am taking the cartesian product here. I have been playing around and came up with:
Let $A,B$ be two f.d. semisimple $k$-algebra. We may define a $k$-algebra $A\times B$. Suppose the class of irreducible reps. of $A$, $B$ are $\{V_i \} , \{W_j \}$ respectively.
Claim: The class of irreducible representations (upto isomorphism) of $A \times B$ consists of $V_i \times \{0 \}$ and $\{0 \} \times W_j$.
My proof: (general idea)
Lemma: Let $M$ be irreducible $A$-module. Then $M$ appears as a composition factor of a composition series of ${_A}A$.
(Using semisimplicity) $A \cong \bigoplus n_i V_i, B \cong \bigoplus m_jW_j$ as $A$-modules, $B$-modules. $n_i, m_j$ are multiplicites. Then $A \times B \cong \bigoplus n_i V_i \times \bigoplus m_j W_j $ as $A \times B$-modules. We thus obtain a composition series for $A \times B$ as $A \times B$-module with composition factors as listed.
Is this correct? Note: I am not sure if finite dimnesional is required.