Remark 8

Let $(B_i)_{i\in I}$ be a family of unital algebras, with $B_i= (A_i, e_i)$ for $i\in I$. Put $A =\prod_iA_i,e= (e_i)_{i\in I}$. Then $(A, e)$ is a unital algebra called the product of the $B_i$. If $x= (x_i)_{i\in I}\in A$, one has Sp$_A(x) =\bigcup_i$ Sp$_{A_i}(x_i)$.