A Ring theory
A ring is a slightly more complex algebraic structure, since it is equipped with two operations, the first of them with the exact same properties of a commutative group operation, whereas the second does not require the existence of inverse.
Definition A.1 A ring42 is a triple \((\mathbb{K},\circ,\ast)\), where \(\mathbb{K}\) is a set and \(\circ\) and \(\ast\) are operations on \(\mathbb{K}\), that is, functions \[\begin{aligned} \circ : \phantom{a} & \mathbb{K}\times \mathbb{K}& \rightarrow & \phantom{a}\mathbb{K}& \qquad\qquad & \ast :& \mathbb{K}\times\mathbb{K}& \rightarrow & \mathbb{K}\\ & (x,y) & \mapsto & \phantom{a} x\circ y, & & & (x,y) & \mapsto & x\ast y, \end{aligned}\] which additionally satisfy the following properties:
\((\mathbb{K},\circ)\) is a commutative group.
\((\mathbb{K},\ast)\) satisfies the following properties:
Associative law: \((x\ast y)\ast z = x\ast(y\ast z)\) for all \(x,y,z\in \mathbb{K}\).
Existence of identity: there exists \(e\in \mathbb{K}\) such that \(e\ast x=x\ast e=x\) for all \(x\in \mathbb{K}\). Such element \(e\) is called the identity element of \(\ast\).
Commutativity: \(x\ast y=y \ast x\) for all \(x,y\in \mathbb{K}\).
Distributive law: \(x\ast(y\circ z)=(x\ast y)\circ(x\ast z)\) for all \(x,y,z\in\mathbb{K}.\)*
Since we have two operations at the same time, we will adopt the additive and multiplicative notation from groups to represent each of them, respectively. That is, we will think of the first operation of a ring as a form of addition and the second as a form of multiplication. On input \(x,y\), we write the result of the first operation by \(x+y\), and the result of the second by \(xy\). The identity elements of each operation are denoted by \(0\) and \(1\), respectively. The inverse of \(x\) with respect to the first operation is denoted by \(-x\). The inverse of \(x\) with respect to the second operation, if exists, is denoted by \(x^{-1}\). The following table summarizes the notation:
| Operation | Operation on input \(x,y\) | Identity element | Inverse of \(x\) |
|---|---|---|---|
| \(+\) | \(x+y\) | \(0\) | \(-x\) |
| \(\cdot\) | \(xy\) | \(1\) | \(x^{-1}\) |
We consider some examples:
\((\mathbb{Z},+,\cdot)\), with usual integer addition and multiplication, is a ring, since \((\mathbb{Z},+)\) is a group with identity \(0\), \((\mathbb{Z},\cdot)\) verifies associativity, commutativity and existence of identity (\(1\)), and it is easy to verify distributivity. Note, however, that \(\mathbb{Z}\) does not contain multiplicative inverses, since for example there is no \(x\in\mathbb{Z}\) such that \(2x=1\). For similar reasons, \((\mathbb{Z}_n,+,\cdot)\), with modular addition and multiplication, is a ring, for \(n\in\mathbb{N}\).
\((\mathbb{Q}[X],+,\cdot)\), where \(\mathbb{Q}[X]\) is the set of polynomials with rational coefficients in variable \(X\), \(+\) is point-wise addition, and \(\cdot\) is point-wise multiplication, is also a ring. The identity elements are the constant polynomials \(p(X)=0\) and \(q(X)=1\), respectively.
Consider the ring \(\mathbb{Z}_9\), with addition and multiplication modulo \(9\). The following table gives the multiplicative inverses of each element:
| \(x\) | \(0\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) |
|---|---|---|---|---|---|---|---|---|---|
| \(x^{-1}\) | \(-\) | \(1\) | \(5\) | \(-\) | \(7\) | \(2\) | \(-\) | \(4\) | \(8\) |
Note that some elements are not invertible, in this case \(0\), \(3\) and \(6\). Those elements in a ring that happen to have a multiplicative inverse receive a special name.
Definition A.2 Let \(\mathbb{K}\) be a ring. An element \(x\in\mathbb{K}\) that has a multiplicative inverse is called a unit. We denote the set of units of \(\mathbb{K}\) by \(\mathbb{K}^*\).
Note that the only thing that \((\mathbb{K},\cdot)\) was missing to be a group is existence of inverses. By restricting ourselves to the subset \(\mathbb{K}^*\) of invertible elements, we have that \((\mathbb{K}^*,\cdot)\) is a group. Note that, in the previous lesson, we defined \(\mathbb{Z}_n^*\) to be the set of invertible elements of \(\mathbb{Z}_n\), so in this new formulation, we can state that \(\mathbb{Z}_n^*\) is actually the set of units of \(\mathbb{Z}_n\). Moreover, we now know that \(\mathbb{Z}_n^*\) is a group with multiplication modulo \(n\).
In light of the discussion above, we start this section by asking ourselves the following question.
Is it possible to have a ring \(\mathbb{K}\) in which all the elements are invertible?
This will never be the case, unless the ring only contains one element, since the additive identity \(0\) cannot have a multiplicative inverse otherwise. To see this, assume that there exists \(x\in\mathbb{K}\) such that \(0\cdot x = 1\). First, \[0x=(1+(-1))x=x+(-x)=0,\] using the distributive law and the properties of the multiplicative identity \(1\). Then we have that \(1=0\). But then, this means that \[x=1x=0x=0,\] using the properties of the multiplicative and additive identities. Thus, we conclude that every element is the same. And rings with just one element are not very interesting.
So, except for the trivial case, we know that \(0\) cannot have an inverse. What about the rest of the elements? In this case, the answer is affirmative: there exist rings in which every non-zero element is invertible, and we call those fields.
Definition A.3 A field is a ring in which every non-zero element has a multiplicative inverse in the ring.
In an abstract algebra context, the definition of a ring is more general, and what we are defining is known as a commutative ring with unity. Nevertheless, we stick to this definition for simplicity, since it is the only type of ring relevant to us.↩︎