Section A.1 Ring Theory
In this class, all rings have a multiplicative identity, written as 1 or \(1_{R}\) is we want to emphasize that we are referring to the ring \(R\text{.}\) This is what some authors call unital rings; since for us all rings are unital, we will omit the adjective. Moreover, we will think of 1 as part of the structure of the ring, and thus require it be preserved by all natural constructions. As such, a subring \(S\) of \(R\) must share the same multiplicative identity with \(R\text{,}\) meaning \(1_{R}=1_{S}\text{.}\) Moreover, any ring homomorphism must preserve the multiplicative identity. To clear any possible confusion, we include below the relevant definitions.
Definition A.1. Ring.
A ring is a set \(R\) equipped with two binary operations, + and \(\cdot\text{,}\) satisfying:
\((R,+)\) is an abelian group with identity element denoted \(0\) or \(0_{R}\text{.}\)
The operation \(\cdot\) is associative, so that \((R, \cdot)\) is a semigroup.
For all \(a, b, c \in R\text{,}\) we have
\begin{equation*}
a \cdot(b+c)=a \cdot b+a \cdot c \quad \text { and } \quad(a+b) \cdot c=a \cdot c+b \cdot c
\end{equation*}
there is a multiplicative identity, written as 1 or \(1_{R}\text{,}\) such that \(1 \neq 0\) and \(1 \cdot a=a=a \cdot 1\) for all \(a \in R\text{.}\)
To simplify notation, we will often drop the \(\cdot\) when writing the multiplication of two elements, so that \(a b\) will mean \(a \cdot b\text{.}\)
Note that the requirement that \(1 \neq 0\) makes it so that the zero ring is not a ring.
Definition A.2. Commutative Ring.
A ring \(R\) is a commutative ring if for all \(a, b \in R\) we have \(a \cdot b=b \cdot a\text{.}\)
Definition A.3. Division Ring.
A ring \(R\) is a division ring if \(1 \neq 0\) and \(R \backslash\{0\}\) is a group under \(\cdot\text{,}\) so every nonzero \(r \in R\) has a multiplicative inverse. A field is a commutative division ring.
Definition A.4. Integral Domain.
A commutative ring \(R\) is a domain, sometimes called an integral domain, if it has no zerodivisors: \(a b=0 \Rightarrow a=0\) or \(b=0\text{.}\) Note that in particular we reserve the word domain for commutative rings.
For some familiar examples, \(\mathrm{M}_{n}(R)\) (the set of \(n \times n\) matrices) is a ring with the usual addition and multiplication of matrices, \(\mathbb{Z}\) and \(\mathbb{Z} / n\) are commutative rings, \(\mathbb{C}\) and \(\mathbb{Q}\) are fields, and the real Hamiltonian quaternion ring \(\mathbb{H}\) is a division ring.
Definition A.5. Ring Homomorphism.
A ring homomorphism is a function \(f: R \rightarrow S\) satisfying the following:
\(f(a+b)=f(a)+f(b)\) for all \(a, b \in R\text{.}\)
\(f(a b)=f(a) f(b)\) for all \(a, b \in R\text{.}\)
\(f\left(1_{R}\right)=1_{S}\text{.}\)
Under this definition, the map \(f: \mathbb{R} \rightarrow \mathrm{M}_{2}(\mathbb{R})\) sending \(a \mapsto\left[\begin{array}{ll}a & 0 \\ 0 & 0\end{array}\right]\) preserves addition and multiplication but not the multiplicative identities, and thus it is not a ring homomorphism.
Exercise A.6.
For any ring \(R\text{,}\) there exists a unique homomorphism \(\mathbb{Z} \rightarrow R\text{.}\)
Definition A.7. Subring.
A subset \(S\) of a ring \(R\) is a subring of \(R\) if it is a ring under the same addition and multiplication operations and \(1_{R}=1_{S}\text{.}\)
So under this definition, \(2 \mathbb{Z}\text{,}\) the set of even integers, is not a subring of \(\mathbb{Z}\text{;}\) in fact, it is not even a ring, since it does not have a multiplicative identity!
Definition A.8. Ideal.
Let \(R\) be a ring. A subset \(I\) of \(R\) is an ideal if:
\(I\) is nonempty.
\((I,+)\) is a subgroup of \((R,+)\text{.}\)
For every \(a \in I\) and every \(r \in R\text{,}\) we have \(r a \in I\) and \(a r \in I\text{.}\)
The final property is often called absorption. A left ideal satisfies only absorption on the left, meaning that we require only that \(r a \in I\) for all \(r \in R\) and \(a \in I\text{.}\) Similarly, a right ideal satisfies only absorption on the right, meaning that \(a r \in I\) for all \(r \in R\) and \(a \in I\text{.}\)
When \(R\) is a commutative ring, the left ideals, right ideals, and ideals over \(R\) are all the same. However, if \(R\) is not commutative, then these can be very different classes.
One key distinction between unital rings and nonunital rings is that if one requires every ring to have a 1 , as we do, then the ideals and subrings of a ring \(R\) are very different creatures. In fact, the only subring of \(R\) that is also an ideal is \(R\) itself. The change lies in what constitutes a subring; notice that nothing has changed in the definition of ideal.
A nontrivial ideal \(I\) of \(R\) is an ideal that \(I \neq R\) and \(I \neq(0)\text{.}\) An ideal \(I\) of \(R\) is a proper ideal if \(I \neq R\text{.}\)
