my very noisey mac fan will not stop

Any recent hardware repairs or upgrades done to this iMac...?

If so, a disconnected or damaged temperature sensor can cause uncontrolled fan speeds.

Run the Apple Hardware test, as per > Use Apple Diagnostics to test your Mac - Apple Support

As the fan system in iMacs (there are 26 years of then out there!) has varied from none to several, we are guessing without knowing the SUB-MODEL of your iMac. That is quickly available by doing "About this Mac..." from your Apple menu:

In the example image, the SUB-MODEL is "(27-inch, Mid 2010)" We need your sub-model to effectively help in this text-based setting. Your symptoms suggest a failed fan motor or a failed temperatrue sensor. Running Apple Hardware Test as suggested will show either problem.

An AHT error starting with "4MOT" is a failed or stuck fan; a error starting with "4SNS" is a sensor problem.

Try: Reset the SMC of your Mac - Apple Support