Question

Let $I\ne R$be an ideal in a commutative ring R with identity. Prove that R/ I is.an integral domain if and only if whenever$ab\in I$ either$a\in I$ or$a\in I$

Short answer

It is proved that R/I isan integral domain if and only if whenever$ab\in I$ either $a\in I$or .$b\in I$

Step-by-step solution

Integral domain:

Integral domain is defined as a non-trivial ring R with identity if it is commutative and contains no divisor of zero, that means if $a,b\in R$and $ab=0$then, either$a=0$or b=0.

Quotient Ring:

Let R be a ring and l be its ideal. Then, set of all cosets of l forms an additive commutative group R/I in which addition is defined by$\left(a+I\right)+\left(b+I\right)=\left(a+b\right)+I$

, for $a,b\in R$and multiplication is defined by$\left(a+I\right)\left(b+I\right)=ab+I$

Proof

Let R/ I be an integral domain, then for all $\left(a+I\right),\left(b+I\right)\in \frac{R}{I}$if $\left(a+I\right)\left(b+I\right)=I$

Then, either $\left(a+I\right)=I$or, $\left(b+I\right)=I$, and we know that $\left(x+I\right)=I$if and only if$x\in I$

So, this is equivalent to saythatwhenever $ab\in I$either $a\in I$or $b\in I$.

Conversely, let $ab\in I$implies either$a\in I$ or $b\in I$.

Now, if $a,b\in I$then we must have, $\left(a+I\right),\left(b+I\right)\in \frac{R}{I}$,

It is given that , $ab\in I$so we can write,

$\left(a+I\right)\left(b+I\right)=ab+I=I$,implies, either $\left(a+I\right)=I$or,$\left(b+I\right)=I$ ,

Which proves that,R/I,is an integral domain.