 # If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative ring with unity. banganX 2020-12-17 Answered
If R is a commutative ring with unity and A is a proper ideal of R, show that $\frac{R}{A}$ is a commutative ring with unity.
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given R is a commutative ring with unity
$⇒ab=ba\mathrm{\forall }a,b\in R$
let $a+A,b+A\in \frac{R}{A}$
$\left(a+A\right)\cdot \left(b+A\right)=ab+A=ba+A=\left(b+A\right)\cdot \left(a+A\right)$
$⇒\frac{R}{A}$ is commutative ring (1)
Now R has unity element rArr there exists $1\in R$ so,
$a\cdot 1=1\cdot a=a\mathrm{\forall }a\in R$
let $a+A\in \frac{R}{A}\mathfrak{1}\in R$ we have $1+A\in \frac{R}{A}:$
we have prove now $1+A$ is the unity element
$\left(a+A\right)\cdot \left(1+A\right)=a\cdot 1+A=a+A$
and $\left(1+A\right)\cdot \left(a+A\right)=1\cdot a+A=a+A\cdot \mathrm{\forall }a+A\in \frac{R}{A}:$
$⇒1+A$ is the unity element in $\frac{R}{A}$ (2)
from (1) and (2) $\frac{R}{A}$ is a commutative ring with unity , hence proved.