Integral Domain Definition

Abstract Algebra · Axiom Academy

Commutative rings with unity and no zero divisors—the foundational structure bridging rings and fields in abstract algebra. In some rings, we can have nonzero elements whose product is zero . These troublesome elements are called zero divisors . 2. The Solution: Integral Domains An integral domain is a ring that explicitly excludes zero divisors, along with ensuring we have the nice properties of commutativity and a multiplicative identity. Commutative: ab = ba for all a, b ∈ R Unity: There exists 1 ∈ R such that 1·a = a·1 = a for all a ∈ R, and 1 ≠ 0 No Zero Divisors: If ab = 0, then a = 0 or b = 0 The absence of zero divisors is equivalent to a very useful property: the cancellation law . Proof idea: If ab = ac and a ≠ 0, then ab - ac = 0, so a(b - c) = 0. Since we have no zero divisors and a ≠ 0, we must have b - c = 0, hence b = c. 4. Examples of Integral Domains Let's look at fundamental examples that show the variety of integral domains: Understanding what isn't an integral domain helps clarify the definition:

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