Example Problems

These problems were sourced from UW Madison MATH 542 Fall 2023 in-class material

1. Show that $0\cdot m=0$ for all $m\in M$. Use this to show that $(-1)\cdot m=-m$. §

Proof. We see that \begin{align*}0\cdot m &= (0 + 0)\cdot m,\\ &=0\cdot m + 0\cdot m.\end{align*} Applying left or right cancellation by $0\cdot m$ yields $0\cdot m=0$. Using this result, \begin{align*}0 &= 0\cdot m,\\ &= (1 + (-1))\cdot m,\\ &=1\cdot m + (-1)\cdot m,\\ &= m + (-1)\cdot m.\end{align*} So $(-1)\cdot m$ is the inverse of $m$ and consequently $(-1)\cdot m=-m$ as desired.

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