# Math Help - Question about quotient rings...

1. ## Question about quotient rings...

If we have a ring, say R and arbitrary ideal I. Is it true that any equation on R, statisfied by elements of R, will be also statisfied in R/I by the cosets, whose representatives are the aforementioned elements. It looks obvious. So obvious, that I'm afraid even to think of doing nice formal proof (I would die to see one though :-).

2. ## Re: Question about quotient rings...

You would have to say what you mean by "equation satisfied by elements of $R$". But yes, there is a natural homomorphism $R \to R/I$, which by definition, preserves sums and products.

$R/I$ doesn't preserve EVERYTHING, though. For example, in $\Bbb Z$ we have:

$2\ast x = 0 \implies x = 0$, but this is NOT true in $\Bbb Z/(6)$ (we might have $x = 3$).

3. ## Re: Question about quotient rings...

10x. The question was somewhat imprecise. The answer, as always, was not