”Independent clause” is really just another name for a simple sentence.

When you join two independent clauses, do one of the following:

  separate them with a semicolon:
      I didn’t want the bananas to spoil; I ate them all that evening.

  separate them with a coordinating conjunction (and, but, for, nor, or, so, yet):
      I didn’t want the bananas to spoil, so I ate them all that evening.

  make them separate sentences:
      I didn’t want the bananas to spoil. I ate them all that evening.

Do not separate two independent clauses with a comma.