# Thread: Proving Rings (sorry for the Tex error)

1. ## Proving Rings (sorry for the Tex error)

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Prove that \left<\frac{R}{I}\,,+\,,\cdot\right> is a ring.
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2. Originally Posted by DanielThrice
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Prove that \left<\frac{R}{I}\,,+\,,\cdot\right> is a ring.
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Perhaps you mean to prove that if $R$ is a conmutative ring, $I\subset R$ ideal of $R$, then $R/I$ is a commutative ring with the standard operations $(x+I)+(y+I)=(x+y)+I$ and $(x+I)(y+I)=xy+I$ . If so, start proving that the sum is well defined that is, if $x+I=x'+I$ and $y+I=y'+I$ then, $(x+I)+(y+I)=(x'+I)+(y'+I)$ .

Fernando Revilla

3. Sorry, (I was half asleep last night) here is what we already know about the problem:

Let R = < R, +, *> (* means multiplication) be a ring, and let I < R be an additive subgroup of < R, + >.
Consider the set of cosets

R/I = {a + I: a is an element of R}

equipped with its own operations + and * defined by

(a + I) + (b + I) = (a + b) + I
(a + I) (b + I) = ab + I

Assume that the operations + and are well-defined <---> the
additive subgroup I satises the following conditions:

ab is an element of I for all a in R and b in I

ba is an element of I for all a in R and b in I