Question

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

Answer 1