We really need to see the pictures.
It sounds like it could be a police luger.
A magazine safety? It may be removed, but the notch they would have cut may still be present.
A sear safety would be big clue, but not all of them had one.
A serial number under the barrel makes me think it was originally a military luger.
The stamps on the right side of the receiver will help confirm that.
BTW, I think you will find the barrel is 10cm.
It is measured from the breech, not from the end of the frame.
(added)
The "AA4" description doesn't sound quite right. It may be an arsenal rework stamp.
The image below has one on the far left and the right. There are many variations.
The Gothic stamps with crowns (c/T c/S c/S) are the usual military acceptance and firing proof stamps.
The 4th eagle with wings up marks it as an Imperial era luger.
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