additive inverse of the zero in a ring
In any ring , the additive identity is unique and usually denoted by . It is called the zero or neutral element of the ring and it satisfies the zero property under multiplication. The additive inverse of the zero must be zero itself. For suppose otherwise: that there is some nonzero so that . For any element we have since is the additive identity. Now, because addition is associative we have
Since is any arbitrary element in the ring, this would imply that (nonzero) is an additive identity, contradicting the uniqueness of the additive identity. And so our suppostition that has a nonzero inverse cannot be true. So the additive inverse of the zero is zero itself. We can write this as , where the sign means “additive inverse”.
Yes, for sure, there are other ways to come to this result, and we encourage you to have a bit of fun describing your own reasons for why the additive inverse of the zero of the ring must be zero itself.
For example, since is the neutral element of the ring this means that . From this it immediately follows that .
|Title||additive inverse of the zero in a ring|
|Date of creation||2013-03-22 15:45:13|
|Last modified on||2013-03-22 15:45:13|
|Last modified by||aplant (12431)|