A quesion about Field of quotients

[wowolala]

The question is to explain why a commutative ring with unity that is not an integral domain cannot be contained in a field.

actually, every field is an integral domain. but above question wants us to compare with thm of Field of quotients..


first , i suppose the R is contained in a field, therefore by thm, can i conlude that R is isomorphic to an integral domain D, moreover, R is an integral domain, so we have contradicton with assumption..


am i right?


can anyone help me ?

thx so much

 

[masnevets]

If you could embed R into a field F, then take a, b in R such that ab = 0 but a and b are nonzero. Then they have the same property in F, which is bad. So your explanation is fine.