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If I is an ideal of R (considering I as an additive subgroup of R), how do the elements of the factor group (R/I) relate with the elements of the factor ring (R/I)?

I know that, If I is an ideal of R, then R/I has a well-defined multiplication: For r, s contained in R, define (r + I)(s + I) = rs + I.

My assumption is that the elements of the factor group (R/I) are equal to the elements of the factor ring (R/I) but I'm not too sure why. Any suggestions would help.

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  • $\begingroup$ You are right. As a set both are same, I mean on the group $R/I$ you are giving some additional structure. $\endgroup$
    – Sumanta
    Commented Nov 12, 2019 at 3:27

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A good way to think about why this is, is that both the quotient group and quotient ring result from treating everything from the ideal like it’s zero, so they should give you the same set.

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