Problem 27 Proof Monday Recitation

27. a. Proposition: Show that every unit of a ring $R$ has a unique inverse.

Proof: Suppose there are inverses $v,w \in R$ of the unit $u \in R$, so $u \cdot v =1$ and $u \cdot w = 1$, so:

(1)
\begin{align} u \cdot v &= u \cdot w \\ v \cdot u \cdot v &= v \cdot u \cdot w \\ 1 \cdot v &= 1 \cdot w \\ v &= w. \end{align}

Thus, there can only be one inverse of an element $u \in R$. $\blacksquare$

b. Proposition: The units of $\mathbb{Z}/n\mathbb{Z}$ are precisely those $[a]$ such that $GCD(a,n)=1$.

Proof: If two elements $a,n$ have $GCD(1,n) = 1$, then there exists a linear combination $b \cdot a + c \cdot n = 1$. By taking the modulo of all elements:

(2)
\begin{align} [b] \cdot [a] + [c] \cdot [n] &= [1] \\ [b] \cdot [a] + [c] \cdot [0] &= [1] \\ [b] \cdot [a] + 0 _{\mathbb{Z}/n\mathbb{Z}}&= [1] \\ [b] \cdot [a] &= [1]. \end{align}

So any elements belonging to $[a]$ have an inverse belonging to $[b]$, making $[a]$ a unit of $\mathbb{Z}/n\mathbb{Z}$. $\blacksquare$

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