Problem 1. Suppose that R is a commutative ring with addition“+” and
multiplication “·”, and that I a subset of R is an ideal in R. Inother words, suppose
that I is a subring of R such that

(x is in I and y is in R) implies x · y is in I.

Define the relation “~” on R by y ~ x if and only if y − x is inI, and assume for the moment
that it is an equivalence relation. Thus we can talk about theequivalence
classes [x] := {y is in R | y ~ x}. Define R/I to be the set ofequivalence classes
R/I := {[x] | x is in R}.
First show that ~ is an equivalence relation so that this all makessense, and
then prove that R/I forms a ring under a suitable addition andmultiplication
operation inherited from R. It should have both an additive and amultiplicative
identity element. The resulting ring R/I is called a “quotientring.”
Specifically, we will want [x] + [y] = [x + y] and [x] · [y] = [x ·y]. Here I
have used the same symbols “+” and “·” to mean two differentoperations (one
operation in R and one in R/I), but unfortunately you’ll have toget used to
that—it’s standard practice to use the same two symbols in everyring.
In any case, showing that these operations satisfy the definitionof a ring
comes down to showing that multiplication is well-defined, and youshould need
to use the fact that i · r is in I whenever i is in I to do so. Theproblem is that [x] is a
single set that can be written multiple ways: if z is in [x], thenfrom the definition
of an equivalence class we can write either [x] or [z] to indicatethe same set.
You have to prove that your definition of addition andmultiplication are the
same regardless of how you write [x] and [y].