Can someone look over what I have done and help me finish my proof? I have no idea how to prove distribution.
Exercise: Suppose G is an abelian group wrt addition and with identity 0. Define a multiplication in G by ab=0 for all a,b in G. Show that G forms a ring wrt these operations.
Since G is abelian wrt multiplication, I would like to use the last two conditions of the following definition.
Definition 5.1b. Alternative Definition of a Ring
Suppose R is a set in which a relation of equality and operation of addition and multiplication are defined. Then R is a ring wrt these operations if these conditions hold:
R forms an abelian group wrt addition.
R is closed wrt and associative multiplication.
Two distributive laws hold in R: x(y + z) = xy + xz, and (x + y)z = xz + yz for all x, y, and z in R.
Proof. Since G is an abelian group with respect to addition, then we only need to prove the last two conditions of the Alternative Definition of a Ring.
Let ab=0 and bc=0 for all a, b, c in G. This means that a(bc)=a(0)=0, and (ab)c=(0)a=0. This shows that G is closed under associative multiplication.
Now we need to show that the distributive laws hold.