here's the definition of an Ideal.

Let R be a commutative ring and let I be a subring of R. Then I is an ideal of R if the following condition holds:

If and then .

so i guess the 'inside' and 'outside' multiplication bit really means which side either the 'a' or the 'r' is on in the above. There's always a few slightly different definitions for things lurking out there so just interpret it how you will.