01 - Ring Structure of Quotient Group

Quotient Ring Construction

Let ideal $I$ in ring $R$

Datum $(R/I,+,\times ,0+I,1+I)$ defines ring structure on $R/I$

Map $q:R\to R/I$ by

$$q(r)=r+I$$

is a surjective ring homomorphism

with $\ker Q=I$

Note that $q:R\to R/I$ is called the quotient homomorphism

Proof - $\ker Q=I$

$$\ker Q=\{r\in R:q(r)=0\}$$

Where $q(r)=0$ means

$$q(r)=r+I+0+I$$

Thus $r\in I$ so $\ker (q)=I$