It is said, that if both a left identity and a right identity in associative ring exists, then it is a two-sided identity as $$e_1=e_1*e_2=e_2$$ Why is associativity required for this property?
2026-03-25 01:15:26.1774401326
Identity element in non-associative ring
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Nobody said associativity was required.
You can search the web and find the phrase "nonassociative ring with unity" used, and see their examples.
In fact, I think you can just take any nonassociative ring $R$ and perform the standard Dorroh construction on $\mathbb Z\times R$ to get a nonassociative ring with identity.