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9.15 Quotient Rings 339
this, however, we need to create the notion of a quotient ring, which we will do in
the next section.
9.15 Quotient Rings
With a group and a normal subgroup we can build a quotient group. In an analogous
way, given a ring and an ideal, we can build a quotient ring. Part of the work
in building a quotient ring is exactly like building a quotient group, so the fact
that a quotient ring has properties R1–R10 has already been done in part. As with
groups, building the quotient structure begins by defining a form of equivalence.
Theorem 9.15.1 Let R be a ring and I an ideal of R. For ring elements a and b,
define a ≡I b if a − b ∈ I. Then ≡I is an equivalence relation on R.
Since R is an abelian group with respect to its addition operation, I is a normal
additive subgroup of R. Therefore, Theorem 9.15.1 is merely a restatement of
Exercise 8.3.2 in its additive form and needs no additional proof. So we are ready
to define the set R/I with its addition and multiplication operations, and show
that it has all properties R1–R10. There are no surprises in the definitions of the
binary operations on R/I. But considering the need for H to be a normal subgroup
of G in showing the binary operation on G/H is well defined, little bells should
be going off in your head as you consider the burden of showing that addition
and multiplication on R/I are well defined. It should be no surprise that you
will exploit the fact that I is an ideal of R in at least part of this demonstration.
What you might not see at first is what kind of ideal I needs to be and where you
will call on the fact that I is this kind of ideal. Since addition in R is commutative,
I is normal as an additive subgroup of R. Thus addition is well defined by our work
in Chapter 8. It’s a different story for multiplication, though. If multiplication is
not commutative, then a left ideal might not be a right ideal, and vice versa. So
we might wonder whether I needs merely to be either a left ideal or a right ideal,
or perhaps both. It turns out that I needs to be a two-sided ideal, and you will see
why when you provide the missing piece of the proof of the following.
Theorem 9.15.2 Let R be a ring and I a two-sided ideal of R. Fora ∈ R write
I + a =[a], where [a] is the equivalence class of a modulo I. Define addition ⊕
and multiplication ⊗ on R/I ={I + a : a ∈ R} by
(I + a) ⊕ (I + b) = I + (a + b) and (I + a) ⊗ (I + b) = I + (ab) (9.70)
Then R/I is a ring under the operations ⊕ and ⊗.
We can talk our way through almost all properties R1–R10, so that your
work in proving Theorem 9.15.2 will be minimal. Since R is an abelian additive
group and I is a normal additive subgroup, R/I has properties R1–R6. Properties
R8–R10 are immediate from the definitions of ⊕ and ⊗. The only property that
takes any real work is R7, showing that ⊗ is well defined, and this is what you