Problem Set - Groups and Monoids

Problem Set - Groups and Monoids

 Let $S$ be a set with an associative operation with identity. Show that the identity is unique. (This tells us that any commutative monoid has only one heart.) Let $S$ be a set with an associative operation with identity. Let $s\in S$ and assume that $s$ has an inverse in $S$. Show that the inverse of $s$ is unique. (This tell us that any element of an abelian group has only one mate.) Let $S$ be a set with identity. Let $s\in S$ and assume that $s$ has an inverse in $S$. Show that the inverse of the inverse of $s$ is equal to $s$. (This tells us that $-\left(-s\right)=s$.) Let $S$ be an abelian group. Show that if $a+c=b+c$ then $a=b$. Let $S$ be a ring. Show that if $s\in S$ then $s\cdot 0=0$.