One can consider quotients of modules. This involves a set of equivalence classes as described
below.

Definition 9.4.1Let A be a D module and let B be a submodule. Then A∕B denotes sets of the forma + B defined by

{a+ b : b ∈ B}

, with the operations defined by

a+ B + (ˆa+ B) ≡ a + ˆa+ B
λ (a+ B) ≡ λa + B

Also, for R a commutative ring and I an ideal, one can define R∕I in the form α + I given by

{α+ λ : λ ∈ I}

. Then the operations are defined as above with I taking the place of B. Also in the case ofthe ring and an ideal, define

(α+ I)(β +I) ≡ αβ +I

The main result about quotient spaces is in the following. It will be reminiscent of what was done with
the field ℤ_{p} for a p a prime.

Proposition 9.4.2A∕B is a D module. Also R∕I is an R module, this for an arbitrary comutativering R. If R is a p.i.d., then R∕

(p)

is a field if p is a prime.

Proof: Note that a + B = â + B is the same as saying that a−â∈ B. Are the operations well defined?
If a + B = â + B and a_{1} + B = a + B,â_{1} + B = â + B, does it follow that

a + B +(ˆa + B) = a1 + B + (ˆa1 + B )

Is

a + ˆa+ B = a1 + ˆa1 +B?

Of course this is so because a + â−

(a1 + ˆa1)

∈ B since B is a group with respect to addition. Next
suppose

a+ B = ˆa + B

Is

λa + B = λˆa+ B?

This is also so because B is a module so λa−λâ = λ

(a − ˆa)

∈ B. The operations are indeed well defined
and so this is indeed a D module.

The claim about R∕I is entirely similar except here one needs to consider the operation of
multiplication. Why is it well defined? Say α + I =

ˆα

+ I and β + I =

ˆβ

+ I. Then α−

αˆ

∈ I and β −

ˆβ

∈ I.
This operation of multiplication is well defined if and only if αβ −

ˆα

ˆβ

is in I. However, this difference
is

( ˆ) ˆ
α β − β + β(α − ˆα) ∈ I

because I is an ideal. Thus the operation is well defined and R∕I is an R module and in addition, it is also
a ring.

Consider the last claim where p is prime. If ρ ∈ R and ρ +

(p)

≠0, this says nothing more than that p
fails to divide ρ. Thus the two are relatively prime and so it follows that there exist σ,τ ∈ R such
that