Define a new operation of addition in Z by x(+)y = x + y -1 with a new multiplication in Z by x(*)y = x+y-xy. Verify that Z forms a ring with respect to these operations.
In order to show it is a ring, I must show that R forms an abelian group with respect to addition, R is closed with respect to an associative multiplication, and two distributive laws hold in R.
How should I start proving those 3 things.
for abelian group: x + y -1 = y + x -1 and x+y-xy = y+x-yx is this right?
for associative : (x+y) - xy = x + ( y - xy ) is this right?
for distributive law: z(x+y-1) = (x+y-1)z and z(x+y-xy) = (x+y-xy)z is this right?