condition on a near ring to be a ring
In short, a distributive near-ring with is a ring.
Before proving this, let us list and prove some general facts about a near ring:
Every near ring has a unique additive identity: if both and are additive identities, then .
If and are additive inverses of , then and . ∎
, since is the (unique) additive inverse of .
There is no ambiguity in defining “subtraction” on a near ring by .
iff , which is just the combination of the above three facts.
If a near ring has a multiplicative identity, then it is unique. The proof is identical to the one given for the first Fact.
If a near ring has a multiplicative identity , then .
. Therefore since has a unique additive inverse. ∎
We are now in the position to prove the theorem.
Set and . Then
Therefore, by Fact 5 above. ∎
|Title||condition on a near ring to be a ring|
|Date of creation||2013-03-22 17:19:54|
|Last modified on||2013-03-22 17:19:54|
|Last modified by||CWoo (3771)|