Abstract algebra/Algebras

Let $R$ be a ring. An $R$-algebra is a pair $(f,S)$ where $f:R\to S$ is a ring homomorphism. By convention, if $R$ is a commutative ring with multiplicative identity, then $f$ should preserve that identity. An $R$-algebra homomorphism $h:(f,S)\to (g,T)$ is a ring homomorphism so that $g\circ h=f$.