and quotient groups extend in the obvious way to R-modules.

De¬nition 14.3. Let M be an R-module. A subset N is a submodule of

M if

(i) N is a subgroup of the additive group M , and

(ii) N is closed under scalar multiplication; that is, for all a ∈ R and

± ∈ N , we have a± ∈ N .

It is easy to see that a submodule N of an R-module M is also an R-

module in its own right, with addition and scalar multiplication operations

inherited from M .

Expanding the above de¬nition, we see that a subset N of M is a sub-

module if and only if for all a ∈ R and all ±, β ∈ N , we have

± + β ∈ N, ’± ∈ N, and a± ∈ N.

Observe that the condition ’± ∈ N is redundant, as it is implied by the

condition a± ∈ N with a = ’1R .

For m ∈ Z, it is easy to see (verify) that not only are mM and M {m}

subgroups of M (see Theorems 8.6 and 8.7), they are also submodules of M .

Moreover, for a ∈ R, aM := {a± : ± ∈ M } and M {a} := {± ∈ M : a± =

0M } are also submodules of M (verify).

302 Modules and vector spaces

Let ±1 , . . . , ±n be elements of M . In general, the subgroup ±1 , . . . , ±n

will not be a submodule of M . Instead, let us consider the set ±1 , . . . , ±n R ,

consisting of all R-linear combinations of ±1 , . . . , ±n , with coe¬cients

taken from R:

:= {a1 ±1 + · · · + an ±n : a1 , . . . , an ∈ R}.

±1 , . . . , ±n R

It is not hard to see (verify) that ±1 , . . . , ±n R is a submodule of M con-

taining ±1 , . . . , ±n ; it is called the submodule spanned or generated by

±1 , . . . , ±n . Moreover, it is easy to see (verify) that any submodule contain-

ing ±1 , . . . , ±n must contain ±1 , . . . , ±n R . As a matter of de¬nition, we

allow n = 0, in which case, the spanned submodule is {0M }.

If N1 and N2 are submodules of M , then N1 + N2 and N1 © N2 are not

only subgroups of M , they are also submodules of M (verify).

Example 14.9. For integer ≥ 0, de¬ne R[X]< to be the set of polynomials

of degree less than . The reader may verify that R[X]< is a submodule of

the R-module R[X]. If = 0, then this submodule is the trivial submodule

{0R }. 2

Example 14.10. Let G be an abelian group. As in Example 14.6, we can

view G as a Z-module in a natural way. Subgroups of G are just the same

thing as submodules of G, and for a1 , . . . , an ∈ G, the subgroup a1 , . . . , an

is the same as the submodule a1 , . . . , an Z . 2

Example 14.11. Any ring R can be viewed as an R-module in the obvious

way, with addition and scalar multiplication de¬ned in terms of the addition

and multiplication operations of R. With respect to this module structure,

ideals of R are just the same thing as submodules of R, and for a1 , . . . , an ∈

R, the ideal (a1 , . . . , an ) is the same as the submodule a1 , . . . , an R . 2

Example 14.12. Let ±1 , . . . , ±n and β1 , . . . , βm be elements of an R-

module. Assume that each ±i can be expressed as an R-linear combination

of β1 , . . . , βm . Then the submodule spanned by ±1 , . . . , ±n is contained in

the submodule spanned by β1 , . . . , βm .

One can see this in a couple of di¬erent ways. First, the assumption that

each ±i can be expressed as an R-linear combination of β1 , . . . , βm means

that the submodule β1 , . . . , βm R contains the elements ±1 , . . . , ±n , and

so by the general properties sketched above, this submodule must contain

±1 , . . . , ±n R .

14.3 Module homomorphisms and isomorphisms 303

One can also see this via an explicit calculation. Suppose that

m

±i = cij βj (i = 1, . . . , n),

j=1

where the cij are elements of R. Then for any element γ in the submodule

spanned by ±1 , . . . , ±n , there exist a1 , . . . , an ∈ R with

n n m m n

γ= ai ±i = ai cij βj = ai cij βj ,

i=1 i=1 j=1 j=1 i=1

and hence γ is contained in the submodule spanned by β1 , . . . , βm . 2

If N is a submodule of M , then in particular, it is also a subgroup of

M , and we can form the quotient group M/N in the usual way (see §8.3).

Moreover, because N is closed under scalar multiplication, we can also de¬ne

a scalar multiplication on M/N in a natural way. Namely, for a ∈ R and

± ∈ M , we de¬ne

a · (± + N ) := (a±) + N.

As usual, one must check that this de¬nition is unambiguous, that is, if

± ≡ ± (mod N ), then a± ≡ a± (mod N ). But this follows from the fact

that N is closed under scalar multiplication (verify). One can also easily

check (verify) that with scalar multiplication de¬ned in this way, M/N is

an R-module; it is called the quotient module of M modulo N .

14.3 Module homomorphisms and isomorphisms

Again, throughout this section, R is a ring. The notion of a group homo-

morphism extends in the obvious way to R-modules.

De¬nition 14.4. Let M and M be modules over R. An R-module ho-

momorphism from M to M is a map ρ : M ’ M , such that

(i) ρ is a group homomorphism from M to M , and

(ii) for all a ∈ R and ± ∈ M , we have ρ(a±) = aρ(±).

An R-module homomorphism is also called an R-linear map. We shall

use this terminology from now on. Expanding the de¬nition, we see that a

map ρ : M ’ M is an R-linear map if and only if ρ(± + β) = ρ(±) + ρ(β)

and ρ(a±) = aρ(±) for all ±, β ∈ M and all a ∈ R.

Since an R-module homomorphism is also a group homomorphism on the

underlying additive groups, all of the statements in Theorem 8.20 apply. In

304 Modules and vector spaces

particular, an R-linear map is injective if and only if the kernel is trivial

(i.e., contains only the zero element). However, in the case of R-module

homomorphisms, we can extend Theorem 8.20, as follows:

Theorem 14.5. Let ρ : M ’ M be an R-linear map.

(i) For any submodule N of M , ρ(N ) is a submodule of M .

(ii) ker(ρ) is a submodule of M .

(iii) For any submodule N of M , ρ’1 (N ) is a submodule of M .

Proof. Exercise. 2

Theorems 8.21, 8.22, and 8.23 have natural R-module analogs:

Theorem 14.6. If ρ : M ’ M and ρ : M ’ M are R-linear maps, then

so is their composition ρ —¦ ρ : M ’ M .

Proof. Exercise. 2

Theorem 14.7. Let ρi : M ’ Mi , for i = 1, . . . , n, be R-linear maps. Then

the map ρ : M ’ M1 — · · · — Mn that sends ± ∈ M to (ρ1 (±), . . . , ρn (±)) is

an R-linear map.

Proof. Exercise. 2

Theorem 14.8. Let ρi : Mi ’ M , for i = 1, . . . , n, be R-linear maps. Then

the map ρ : M1 — · · · — Mn ’ M that sends (±1 , . . . , ±n ) to ρ1 (±1 ) + · · · +

ρn (±n ) is an R-linear map.

Proof. Exercise. 2

If an R-linear map ρ : M ’ M is bijective, then it is called an R-module

isomorphism of M with M . If such an R-module isomorphism ρ exists,

we say that M is isomorphic to M , and write M ∼ M . Moreover, if

=

M = M , then ρ is called an R-module automorphism on M .

Analogous to Theorem 8.24, we have:

Theorem 14.9. If ρ is a R-module isomorphism of M with M , then the

inverse function ρ’1 is an R-module isomorphism of M with M .

Proof. Exercise. 2

Theorems 8.25, 8.26, 8.27, and 8.28 generalize immediately to R-modules:

Theorem 14.10. If N is a submodule of an R-module M , then the natural

map ρ : M ’ M/N given by ρ(±) = ± + N is a surjective R-linear map

whose kernel is N .

14.3 Module homomorphisms and isomorphisms 305

Proof. Exercise. 2

Theorem 14.11. Let ρ be an R-linear map from M into M . Then the map

ρ : M/ ker(ρ) ’ img(ρ) that sends the coset ± + ker(ρ) for ± ∈ M to ρ(±) is

¯

unambiguously de¬ned and is an R-module isomorphism of M/ ker(ρ) with

img(ρ).

Proof. Exercise. 2