Editing Associative algebra (section)

== Algebra homomorphisms ==
{{main|algebra homomorphism}}
A [[homomorphism]] between two ''R''-algebras is an [[module homomorphism|''R''-linear]] [[ring homomorphism]]. Explicitly, {{nowrap|''φ'' : ''A''<sub>1</sub> → ''A''<sub>2</sub>}} is an '''associative algebra homomorphism''' if
: <math>\begin{align}
  \varphi(r \cdot x) &= r \cdot \varphi(x) \\
      \varphi(x + y) &= \varphi(x) + \varphi(y) \\
         \varphi(xy) &= \varphi(x)\varphi(y) \\
          \varphi(1) &= 1
\end{align}</math>

The class of all ''R''-algebras together with algebra homomorphisms between them form a [[category (mathematics)|category]], sometimes denoted '''''R''-Alg'''.

The [[subcategory]] of commutative ''R''-algebras can be characterized as the [[coslice category]] ''R''/'''CRing''' where '''CRing''' is the [[category of commutative rings]].