Questions 26 & 27 PreWrite

26. a. Prove that equality modulo an ideal I of a ring R is an equivalence relation.
Proof: (Reflexivity)

\begin{align} \forall x \sim _I x \end{align}
\begin{align} x-x \in I \end{align}
\begin{equation} x-x=0 \end{equation}

A subring by definition is nonempty and any number times 0 is 0 so $0 \in I$

\begin{align} x\sim _Iy \rightarrow y \sim _Ix \end{align}
\begin{align} x-y \in I \rightarrow y-x \in I \end{align}
\begin{align} 0 \in I \end{align}
\begin{equation} 0 - (x-y) = y-x \end{equation}

0 is in the subring and it is closed under subtraction.

\begin{align} x \sim _Iy, y \sim _Iz \rightarrow x \sim _Iz \end{align}
\begin{align} x-y \in I, y-z \in I \rightarrow x-z \in I \end{align}
\begin{equation} (x-y) + (y-z) = x-y+y-z = x-z \end{equation}

The subring is closed under addition.

27. a. Show that every unit of a ring R has a unique inverse.
Proof: Suppose for some $x, a, b \in R$ the inverse of x, c is equal to $c = xa, c = xb$.
$cb = b(xa) = a(bx) = ac$
Thus, $cb = ac$ and $b = a$.

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