additive inverse of a sum in a ring
Let be a ring with elements . Suppose we want to find the inverse of the element . (Note that we call the element the sum of and .) So we want the unique element so that . Actually, let’s put where is the additive inverse of and is the additive inverse of . Because addition in the ring is both associative and commutative we see that
since is the additive inverse of and is the additive inverse of . Since additive inverses are unique this means that the additive inverse of must be . We write this as
It is important to note that we cannot just distribute the minus sign across the sum because this would imply that which is not the case if our ring is not with unity.
|Title||additive inverse of a sum in a ring|
|Date of creation||2013-03-22 15:45:02|
|Last modified on||2013-03-22 15:45:02|
|Last modified by||rspuzio (6075)|