“Morally” here is a bit of mathematical slang—it means intuitively.
The additive structure is the addition, the multiplicative structure is the multiplication.
A structure over a set is a collection of relations between elements of the set (2+3=5 is a relation between 2, 3, and 5). The point is that we can abstract away everything until we are left with the basic properties of a structure—for the addition those are the associativity (2+3)+5=2+(3+5), the neutral element 3+0=3, the existence of symmetric elements 3+(-3)=0, and the commutativity 3+2=2+3. Any structure with those properties tends to be called an addition, regardless of the set. And the first three properties defines the group structure.
Now we can do the same thing for the multiplication (which for the integers has the same basic properties, except the existence of a symmetric). And then we can wonder how these structures interact—for example, a*(b+c)=ab+ac, this is called the distributivity of the multiplication over the addition.
“Morally” here is a bit of mathematical slang—it means intuitively.
The additive structure is the addition, the multiplicative structure is the multiplication.
A structure over a set is a collection of relations between elements of the set (2+3=5 is a relation between 2, 3, and 5). The point is that we can abstract away everything until we are left with the basic properties of a structure—for the addition those are the associativity (2+3)+5=2+(3+5), the neutral element 3+0=3, the existence of symmetric elements 3+(-3)=0, and the commutativity 3+2=2+3. Any structure with those properties tends to be called an addition, regardless of the set. And the first three properties defines the group structure.
Now we can do the same thing for the multiplication (which for the integers has the same basic properties, except the existence of a symmetric). And then we can wonder how these structures interact—for example, a*(b+c)=ab+ac, this is called the distributivity of the multiplication over the addition.