TITLE: Are monomorphisms of rings injective?
QUESTION [6 upvotes]: Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?

REPLY [2 votes]: Yes. Suppose $f$ is a monomorphism which is not injective, ie $f(a)=f(b)$. Now consider $g_1:\mathbb{Z}[x]\to R$ given by sending $x\mapsto a$ and $g_2:\mathbb{Z}[x]\to R$ given by $x\mapsto b$. Then $f\circ g_1=f\circ g_2$ despite the two maps being different. Thus monomorphisms must be injective.