The Ring of Z/2Z

Recall from the Rings page that if $+$ and $*$ are binary operations on the set $R$, then $R$ is called a ring under $+$ and $*$ denoted $(R, +, *)$ when the following are satisfied:

• 1. For all $a, b \in R$ we have that $(a + b \in R)$ (Closure under $+$).

• 2. For all $a, b, c \in R$, $a + (b + c) = (a + b) + c$ (Associativity of elements in $R$ under $+$).

• 3. There exists an $0 \in R$ such that for all $a \in R$ we have that $a + 0 = a$ and $0 + a = a$ (The existence of an identity element $0$ of $R$ under $+$).

• 4. For all $a \in R$ there exists a $-a \in R$ such that $a + (-a) = 0$ and $(-a) + a = 0$ (The existence of inverses for each element in $R$ under $+$).

• 5. For all $a, b \in R$ we have that $a + b = b + a$ (Commutativity of elements in $R$ under $+$).

• 6. For all $a, b \in R$ we have that $a * b = b * a$ (Closure under $*$).

• 7. For all $a, b, c \in R$, $a * (b * c) = (a * b) * c$ (Associativity of elements in $R$ under $*$).

• 8. There exists a $1 \in R$ such that for all $a \in R$ we have that $a * 1 = a$ and $1 * a = a$ (The existence of an identity element $1$ of $R$ under $*$).

• 9. For all $a, b, c \in R$ we have that $a * (b + c) = (a * b) + (b * c)$ and $(a + b) * c = (a * c) + (b * c)$ (Distributivity of $*$ over $+$).

We will now look at the ring of $\mathbb{Z} / 2\mathbb{Z}$. Consider the set of integers $\mathbb{Z}$. Partition the set of integers into the set of even integers denoted $[0]_1$ and the set of odd integers $[1]_2$. The notation $[0]_2$ corresponds to the set of integers $z \in \mathbb{Z}$ such that $z \equiv 0 \pmod 2$ (i.e., even integers) and the notation $[1]_2$ corresponds to the set of integers $z \in \mathbb{Z}$ such that $z \equiv 1 \pmod 2$ (i.e., odd integers). Therefore $\mathbb{Z} = [0]_2 \cup [1]_2$ and $[0]_2 \cap [1]_2 = \emptyset$.

Define the set $\mathbb{Z} / 2 \mathbb{Z}$ as the set of sets:

(1)

\begin{align} \quad \mathbb{Z} / 2 \mathbb{Z} = \{ [0]_2, [1]_2 \} \end{align}

Let $+$ be the operation of addition and let $*$ be the operation of multiplication.

We know that an even + even = even, even + odd = odd, odd + even = odd, and odd + odd = even. Therefore we can represent the addition on $\mathbb{Z} / 2 \mathbb{Z}$ by the following table:

We also know that an even * even = even, even * odd = even, odd * even = even, and odd * odd = odd. Therefore we can represent the multiplication on $\mathbb{Z} / 2 \mathbb{Z}$ by the following table:

If we define $+$ and $*$ in the manner described in the two tables above, then we can verify that $(\mathbb{Z} / 2 \mathbb{Z}, +, *)$ is a ring.

Clearly the sum of any two integers is either even or odd, so $\mathbb{Z} / 2 \mathbb{Z}$ is closed under addition. We also know that the addition of integers is associative implying that the sign of any sum of integers will be the same regardless of the order in which addition occurs, so $+$ is associative. The identity elements in $\mathbb{Z} / 2 \mathbb{Z}$ is $[0]_2$ since adding an even integer to an even integer is still an even integer, while adding an even integer to an odd integer is still and odd integer. We have that $[0]_2$ is the additive inverse of $[0]_2$ since an even number plus an even number gives us our additive identity of even numbers. We also have that $[1]_2$ is the additive inverse of $[1]_2$ since an odd number plus an odd number gives us our additive identity of even numbers. Furthermore, we know that the sum of integers is commutative implying the sign of any rearrangement of the summands will be the same, so $+$ is commutative.

The other axioms regarding $*$ on $\mathbb{Z} / 2\mathbb{Z}$ can be verified similarly to show that $(\mathbb{Z} / 2\mathbb{Z}, +, *)$ is a ring.

The general structure of $\mathbb{Z} / 2 \mathbb{Z}$ can also be described with modular arithmetic. Consider the set $R = \{ 0, 1 \}$. For all $x, y \in R$ define $+$ and $*$ by:

(2)

\begin{align} \quad x + y = (x + y) \pmod 2 \quad \mathrm{and} \quad x * y = (x \cdot y) \pmod 2 \end{align}

Note that the $+$ and $\cdot$ on the righthand side of each equation above denotes standard addition and standard multiplication respectively.

It's not hard to verify that $(R, +, *)$ is essentially the exact same ring as $(\mathbb{Z} / 2 \mathbb{Z}, +, *)$ - even the addition and multiplication tables above are identical as the reader should verify.