Let 〈 R , + , • 〉 be a commutative ring. A nonempty subset I in R is called an ideal if I is closed under addition and under inside-outside multiplication.
what is meant by "inside-outside multiplication"
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 $\displaystyle a \in I$ and $\displaystyle r \in R$ then $\displaystyle r \cdot a = a \cdot r \in I$.
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.
I am not familiar with that term, but it could also be referring to the fact that ideals are sort of super closed under multiplication.
What I mean is $\displaystyle I \subset R$, so there could be things in the ring R that are not in the ideal I, "outside" the ideal if you will. In an ideal, when you multiply an element of the ideal by any element in the ring, it stays "inside" the ideal. $\displaystyle ri \in I$ for all $\displaystyle r\in R$ and for all $\displaystyle i \in I$.
This could possibly be what they mean by being closed under inside-outside multiplication, but I do not know for sure.
Of further note, a ring need not be commutative to have an ideal, the definition given above might be slightly misleading the way it is worded. The definition you have, says you are given a commutative ring first, and then it defines the ideal in commutative rings, but there are certainly ideals in non-commutative rings, you just need to specify left and right ideals, kind of like left and right cosets for groups.
An example from wikipedia (http://en.wikipedia.org/wiki/Ideal_(ring_theory))
"The set of all n-by-n matrices whose last column is zero forms a left ideal in the ring of all n-by-n matrices. It is not a right ideal. The set of all n-by-n matrices whose last row is zero forms a right ideal but not a left ideal."