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Handout Wednesday, 9/3
The homework for today asked you to explore two operations. Are there questions about either of these?
Many of our usual sets of numbers have two operations that work with them. The standard we will adopt when working with such sets is to call one of the operations addition and the other multiplication. We will write the operation like these are normally written. But notice that that doesn’t mean that addition and multiplication are always the “normal” operations.
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What makes the operations count as abstract versions of addition and multiplication? First, we require that the addition is associative, commutative, there is an additive identity, and every element has an additive inverse. In other words, the set with addition forms an abelian group.
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The multiplication operation can also have some or all of these properties (for inverses, we restrict to only the non-zero elements, where zero is the name of the additive identity). When the multiplication operation has all four properties, we call the set with the two operations a field.
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There is one more important property that we must have: some way that addition and multiplication interact. It turns out that all we need to require is the distributive property: for all \(a\text{,}\) \(b\text{,}\) and \(c\) in the set, \(a (b + c) = a b + a c\) and \((a + b) c = a c + b c\text{.}\)
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What about all the other properties of fields that are also true? Like that \(0a = 0\) for all \(a\) in the field? Well, this follows from the distributive property and the fact that \(0\) is the additive identity. We will consider more of these facts later.
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Common examples of fields include \(\Q\text{,}\) \(\R\text{,}\) and \(\C\text{.}\) But also \(\Z_5\text{.}\) Not \(\Z_6\) (why?).
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What about \(\Z\text{?}\) We are missing just one property: multiplicative inverses. We say that \(\Z\) is an example of an integral domain. What exactly that means will come later.
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In general, we can drop one or more of the properties for multiplication and get an interesting structure. We always want multiplication to be associative and distributive over addition. When we have at least this, we call the structure a ring.
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Add a multiplicative identity, we get a ring with unity or ring with identity. Add commutativity for multiplication, and we get a commutative ring. Add both: commutative ring with unity. If we add inverses, but maybe not commutativity, we get a division ring.
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An integral domain is a commutative ring with unity that also almost has inverses. That’s not to say it has inverses for many elements. In fact, there is an integral domain in which only two elements have inverses (the integers, with \(1\) and \(-1\)). What does “almost inverses” mean here?
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Consider the integers and an equation like \(3x = 12\text{.}\) Can you solve for \(x\text{?}\) We obviously know that \(x = 4\text{,}\) and in some ways we can think of dividing both sides by \(3\) to find this. But what does it mean to divide by \(3\) in the integers? We don’t have a multiplicative inverse for \(3\) in the integers.
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What we do have is the cancellation property. This means that if \(ax = ay\) for some \(a\text{,}\) \(x\text{,}\) and \(y\text{,}\) and \(a\) is not zero, then \(x = y\text{.}\) In other words, we can "cancel" the \(a\) from both sides of the equation, even though we don’t have \(a^{-1}\) to multiply both sides by.
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For now, think of an integral domain as a commutative ring with unity that has the cancellation property. Technically we will define it differently later, but in an equivalent way.
