Q: The ring {0, 2, 4 , 6, 8] under addition and multiplication modulo 10 has a unity. Find it.
A: By definition we know that the unity of a ring is a nonzero element that is an identity under multiplication (of the ring). In other words any element of the ring multiplied by the identity will give us the original element.
So we can test each element of the ring {0, 2, 4 , 6, 8] to see which element satisfies these conditions.
2*2 = 4 4 modulo 10 = 4 <> 2
So 2 can not a unity.
4*4 = 16 16 modulo 10 = 6 <> 4
So 4 can not be a unity.
6*6 = 36 36 modulo 10 = 6 = 6
Let's test it on the other elements of the ring:
2*6 = 12 12 modulo 10 = 2 = 2
4*6 = 24 24 modulo 10 = 4 = 4
8*6 = 48 48 modulo 10 = 8 = 8
even 0*6 = 0 0 modulo 10 = 0 = 0
Since for every belonging to the ring we have that a*6 = a mod 10 we have that 6 is the unity. Moreover by thereom 12.2 this unity is unique so 8 cannot be a unity.
