## Definition ## Given a [[ring]] $R$, an __$R$-algebra__ is a [[ring]] $A$ with a [[ring homomorphism]] $f:R \to A$. ## Properties ## If $R$ is a [[commutative ring]] and the $R$-algebra $A$ has a [[commutative ring homomorphism]] $g:R \to Z(A)$ into the [[center]] $Z(A) \subseteq A$ of $A$, as well as a term $a: i \circ g = f$, where $i:Z(A) \subseteq A$ is the associated [[monic function]] for subtype $Z(A)$ of $A$, then $A$ is an [[associative algebra|associative]] [[unital algebra|unital]] [[algebra (module theory)|algebra]] in [[module]] theory. ## See also ## * [[ring]] * [[commutative ring]] * [[commutative algebra (ring theory)]] * [[algebra (module theory)]]