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

How do we prove that the operations + and are well-defined

You have to probe that

.

I don't understand what you meant with the following part below

Tonio
<---> 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

?