One thing we can be sure of: the concept of left and right ideals is applicable only for non commutitive rings. In a commutative ring such as a modulus ring, left and right ideals are the same, since x times r and r times x are not different. So any time they start tlking about left and right ideals or splitting, you know they are talking about some non commutitive object.
Everywhere I turn are statements I do not understand. All I can do is take them one at a time, putting them on the hold list until I can get to them. For instance, why and how there are exactly 21 different quadratic fields is still quite a mystery to me.