Quotient rings link to first isomorphism theorem

Question

Today in class someone told me that quotient Ring being a domain or a quotient Ring being a field is linked to the first isomorphism theorem. I personally cannot see this link and thought it was just a result of two different theorems. Am I wrong or is there a connection between these topic area?

Answer 1

The deal is this - if the ideal you are quotienting out is a prime ideal, then the quotient ring is an integral domain. It the ideal you are quotienting out is a maximal ideal, then the quotient ring is a field. (This is all assuming you’re starting with a commutative ring with unity, of course.)

I don’t believe any use of the First Isomorphism Theorem (FIT) is made in the proofs of these facts, but it can go hand in hand with them to show that some other ring is in fact an integral domain or a field, by exhibiting a homomorphism onto that ring whose kernel is a prime or maximal ideal. The FIT then shows that the image of that homomorphism is a field/ID since it is isomorphic to a quotient ring which is a field/ID.

Answer 2

In quatient ring it all depends upon the kernel.Kernel is actually an ideal.If your kernel is any prime ideal then quatient ring is integral domain but if the kernel is maximal ideal then quatient ring will be a field.