| 1. | A regular ring is reduced but need not be an integral domain.
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| 2. | The proof uses induction so it does not apply to all integral domains.
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| 3. | Notice also that the unit in that integral domain.
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| 4. | A B�zout domain is an integral domain in which B�zout's identity holds.
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| 5. | This shows that integral domains and division rings don't have such idempotents.
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| 6. | Rational expressions are the quotient field of the polynomials ( over some integral domain ).
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| 7. | If however R is not an integral domain, then the conclusion need not hold.
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| 8. | First adjoin negatives to get an integral domain, then take its field of fractions.
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| 9. | "Integral domain " is defined almost universally as above, but there is some variation.
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| 10. | Noncommutative integral domains are sometimes admitted.
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