Is this alternator with or without a rectifier

With. It's got round diodes in that inside plate.
It's not Cooper S. Is later.

It'd be helpful if Lucas hadn't used 15AC on 2 different alternators.
Edit
It has an internal regulator. Was used on Clubman.
The UK made? earlier 15AC was the one with plastic half end cover, and external regulator, as fitted to Mini K, MK 2 S, and MK 2 Mini Matic.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

I had one of those on my car, it was dated on the back and stamped with the Lucas part number 62921219A

viewtopic.php?f=2&t=92267

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Yes it has a rectifier but you want to know if it has a regulator built in which it does. If external regulation was required there would be a terminal in the slot marked "f".
As previously stated there were two versions of this 15AC, the one you hve and one which required an external reg having an extra terminal at the F slot.
On a slightly different tack yours in only a 29 amp output. You can buy an alternator with 50A output if you need more power.
Hope this helps

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Cheers Dave.