Given a ring$R$ and a two-sided ideal$I$ with canonical $R$-$R$-bimodule monomorphism$i:I \hookrightarrow R$, the quotient of $R$ by $I$ is the initial two-sided $R$-algebra$R/I$ with canonical ring homomorphism$h:R \to R/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-algebra $S$ with canonical ring homomorphism $k:R \to S$ such that for every element $a \in I$, $k(i(a)) = 0_S$, there is a unique ring homomorphism $l:R/I \to S$ such that $l \circ h = k$.

Quotient by left ideals

Given a ring$R$ and a left ideal$I$ with canonical left $R$-modulemonomorphism$i:I \hookrightarrow R$, the quotient of $R$ by $I$ is the initial left $R$-algebra$R/I$ with canonical ring homomorphism$h:R \to R/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-algebra $S$ with canonical ring homomorphism $k:R \to S$ such that for every element $a \in I$, $k(i(a)) = 0_S$, there is a unique ring homomorphism $l:R/I \to S$ such that $l \circ h = k$.

Quotient by right ideals

Given a ring$R$ and a right ideal$I$ with canonical right $R$-modulemonomorphism$i:I \hookrightarrow R$, the quotient of $R$ by $I$ is the initial right $R$-algebra$R/I$ with canonical ring homomorphism$h:R \to R/I$ such that for every element $a \in I$, $h(i(a)) = 0$: for any other $R$-algebra $S$ with canonical ring homomorphism $k:R \to S$ such that for every element $a \in I$, $k(i(a)) = 0_S$, there is a unique ring homomorphism $l:R/I \to S$ such that $l \circ h = k$.