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Handout Friday 10/3
We explored more of the activity we started the previous time and concluded that in some rings, even if not every element has a multiplicative inverse (i.e., even if the ring is not a field), we can still cancel. In particular.
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Some rings have the cancellation property: For all elements \(a, b, c\) in the ring with \(a \ne 0\text{,}\) if \(ab = ac\) then \(b=c\text{.}\)
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This is not true in rings like \(\Z_6\) though, since \(2\cdot 1 = 2 \cdot 4\text{,}\) for example.
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We also noticed that some rings have the zero product property: For all elements \(a, b\) in the ring, if \(ab = 0\) then either \(a=0\) or \(b=0\text{.}\)
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Definitely all fields have both these properties. But also \(\Z\) and \(R[x]\) (as long as \(R\) has the properties).
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We proved that the two properties are in fact equivalent. This used the observation that \(ab = ac\) if and only if \(a(b-c) = 0\text{.}\)
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We call rings with these properties integral domains.
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Another way to describe integral domains are as commutative rings with unity that have no zero divisors, i.e., no element \(a \ne 0\) such that some \(b \ne 0\) exists with \(ab = 0\text{.}\) In other words, a zero divisor is a counterexample to the zero product property, so a ring with no zero divisors is a ring in which no elements disobey the zero-product property.
