Let $(R,+,*)$ be an integral domain such that $a*a =a$ for all $a\in R$. What will be the size of $|R|$?
My attempt : $|R| = 1$, as $a*a=a$ by cancellation law $a = e$ which means $|R|=1$
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2(1) In an integral domain the cancellation need only hold for nonzero elements. (2) In the ring with one element, by definition that element is zero, and usually the zero ring is by definition excluded from being an integral domain. – Travis Willse Apr 18 '18 at 09:17
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Is my proof correct? – abaa Apr 18 '18 at 10:34
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No. It is not correct. As Travis explained cancellation does not work when $a=0$. – Jyrki Lahtonen Apr 18 '18 at 17:35