I'm reading Theorem 8.4 and its proof from textbook Analysis I by Amann/Escher.
From this Wikipedia page, I got
From this Wikipedia page, semiring mentioned above is defined to have a multiplicative identity.
So both my textbook and Wikipedia assert that the binomial theorem holds for a ring with unity. On the other hand, I do not see that the proof in my textbook utilizes this property.
My question:
Does the binomial theorem hold for a ring without unity?
Best Answer
The right-hand side of the equation $$(a+b)^n = \sum_{k=0}^n \binom nk a^k b^{n-k}$$ is not well defined in a ring without unity because $a^0$ and $b^0$ are not defined. You can claim instead that $$(a+b)^n = a^n + b^n + \sum_{k=1}^{n-1} \binom nk a^k b^{n-k}$$ which can be obtained either by the same proof, or by simply expanding the product and counting the terms as @Captain Lama suggested.