Valuation theory on semisimple algebras used in the paper of Cohen-Martinet: reference request

Question

I'm currently reading the paper of Henri Cohen & Jacques Martinet "Etude heuristique des groupes de classes des corps de nombres"

On the 2nd section, they recall some facts on valuations, completions of semisimple algebra over $\mathbb{Q}$.

They refered the book "Algebren" by Max Deuring, however unfortunately, I can't read German.

I could find some reference on the valuations on central simple algebras over a field. But I couldn't find any on general semisimple algebras.

If anyone can suggest a good reference useful for reading the paper, that would be a great help to me.

Answer 1

I'm sorry for not giving enough thought.

Any prime ideal of $A=A_{1}\times A_{2} \times \cdots \times A_{n}$ contains all but one $1 \times \cdots A_{i} \cdots \times 1$ ($e_{i}e_{j}=0$).

Any maximal ideal is of the form $A_{1} \times \cdots m_{i} \cdots A_{n}$ for some $i$ and a maximal ideal $m_{i}$.

Using the definition of completion that uses the Cauchy sequence, any sequence from $A_{1} \times \cdots \{0\} \times \cdots \times A_{n}$ is a null-sequence.

Hence the completion is the completion of $A_{i}$ with respect to $m_{i}$.

Although this question was elementary, I guess it's proper to let this remain undeleted.

Thank you.