A, sulfur bacteria can use soluble or dissolved sulfur-containing compounds in the growth process to obtain energy, so as to survive, and do not need to obtain it from organic substances manufactured by producers, so they belong to producers. Therefore, the option is correct;

B, nitrifying bacteria can use the energy released by oxidizing some substances to synthesize organic matter from carbon dioxide and water, which belongs to producers. Therefore, the option is correct;

C. Mushrooms are fungi that survive by decomposing decaying organic matter and belong to decomposers. Therefore, the option is incorrect;

D. Auricularia auricula is a fungus that lives by decomposing decaying organic matter and belongs to decomposers. Therefore, the option is incorrect.

So choose a and B.