Actually the opposite of this. When you receive an O2 code for "too rich", it means that the FUEL content (un-burned) is higher than parameters set in the ECU to signal the condition.

I don't work much on the BMWs that I own. I'm older, not as "in need of performance" as I used to be, have enough modified (and occasionally) headaches already, and they're decent enough performing for my needs as "everyday cars". I have customers that I do minor work for one their BMWs (suspension, exhaust, intake, light electronics, etc.), but I'm not building engines/transmissions, nor doing hard coding on the DMEs.

I note that as a basis for my knowledge level on these cars, as is not as extensive as with other marques, yet a majority of the knowledge transfers over (some particulars as in BMWs DME code are where I'm not professing expertise).

I would research these two facts, but I'm pretty confident in them being correct.

1) The DME will reset both short-term, and long-term fuel trims (learned values in fuel adjustment from the base fuel mapping) if the power is removed (disconnect a battery, wait a little bit for the capacitors in the car to leak down voltage, then re-connect).

2) The E53s still use a 1 volt, narrow band sensor (as opposed to the newer cars with 5V wide band sensors).

That said, if you have a sacan tool that will allow you to read sensor values, I would warm the car up to operating temperature, shut it off, and reset the DME via the battery disconnect. Wait a few minutes for the capacitors to leak down, reconnect, start it, and read both front O2 sensor readings with your scan tool (Bank 1, Sensor 1 & Bank 2, Sensor 1---may be listed just as "B1S1 & B2S1" on your particular scan tool). Watch the values for a minute or two, and see if they're way off from one another in voltage (0.00 volts to 1.00 volts).

Then allow it to idle for 10-15 minutes to see what the Long Term Fuel Trim (or learned fuel trim percentage--the phrasing can differ per scan tool used) is, and then check the Short Term Fuel Trim (STFT) to see if it has settled down into the +/- 8% range.

This will give you a baseline to start with.

As others have mentioned, vacuum leaks, etc., can cause issues, however, that will typically cause a condition shown on BOTH banks at that point. If you can find the TSB on a working page that the gentleman above pointed out, I would also research that.

The above said, the "typical" things to cause one bank to be rich, and the other fine (using your baseline values above for comparison) are:

1) A bad/fouling spark plug.

2) Weak Coil on one/multiple cylinders (I've already dealt with this on my 60K mile 4.4)

3) Dirty/faulty injector. The spray pattern can go from "Mist" to "Stream" with very little dirt, and/or have the the injector never completely "seal" between pulses.

4) Valve seal leaking, and introducing oil into the cylinder

5) Weak compression in a cylinder/cylinders on that bank (valve seats, rings, etc.).

You'd probably notice a decent "oil burning" smell on those last two.

It's good that you swapped the O2s to start with, ruling out a bad sensor.

It's all just a matter of diagnostics/ruling out what *isn't* wrong to narrow the possibilities.

Do the easy stuff first.

Josh