00:01
All right, so let x be a non -zero.
00:11
Well, let's let x be a zero divisor in a commutative ring with unity.
00:41
Then x is not zero.
00:45
And there is non -zero y for which xy is equal to yx is equal to zero.
01:09
Suppose, for the sake of contradiction, that x is a unit, then there exists x inverse such that x inverse times x equals x, x inverse is equal to 1.
01:54
But we'll see that there's a problem here because if xy equals zero, and we said that y was non -zero, then we just multiply x inverse on both sides.
02:06
Then that gives us 1 times y is equal to 0, so y is equal to 0, which is a contradiction.
02:32
We conclude that a zero divisor in a community of ring with unity cannot be a unit, which is what we wanted to prove.
03:17
For your next part, not a zero divisor and not a unit, it actually depends on the communitive ring with unity.
03:35
For finite, commutative rings with unity, an element, an element, is either a unit or a zero divisor...