Abstract Algebra/Algebras

Let R be a ring. An R-algebra is a pair $(f,\,S)$ where $f\; \colon\; R\;\to\; S$ 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 $g\circ h=f$.