M-302: RINGS AND MODULES
ASSIGNMENT - III
Total Marks: 30.
Due Date: November 29, 2020, 22:00 hrs.
All rings are non-zero rings with identity.
1. Let R be a commutative ring and M be an R-module. Prove that
HomR (R, M ) ∼
= M as R-modules. [4]
2. Let R be a ring, and let M1 , M2 , · · · , Ms be R-modules. Let Ni be a
submodule of Mi , for each i = 1, 2, · · · , s. Show that ⊕si=1 Ni is a submodule
of ⊕si=1 Mi and
⊕si=1 Mi / ⊕si=1 Ni ∼ = ⊕si=1 Mi /Ni
as R-modules. [4]
3. Let R be a ring, and let I be a left ideal of R. Then for every positive
integer n, show that
Rn /IRn ∼
= R/IR ⊕ · · · ⊕ R/IR (n − times)
as R-modules. Conclude that, if R is commutative then Rm ∼
= Rn (as R-
modules) if and only if m = n. [2 + 3 = 5]
4. Let I be a nilpotent ideal in a commutative ring R i.e. I n = (0)
for some integer n ≥ 1, M, N be R-modules and let φ : M −→ N be an
R-homomorphism. Show that if the induced map φ : M/IM −→ N/IN is
surjective, then φ is surjective. [4]
5. Let R be a commutative ring.
(i) Let M1 , M2 , · · · , Mt , N be R-modules. Then there is an R-module
isomorphism
HomR ⊕ti=1 Mi , N ∼ = ⊕ti=1 HomR (Mi , N ).
(ii) Let M, N1 , N2 , · · · , Ns be R-modules. Then there is an R-module
isomorphism
HomR M, ⊕si=1 Ni ∼ = ⊕si=1 HomR (M, Ni ).
(iii) Let F be a free R-module of finite rank n, and M be an R-module.
Prove that HomR (F, M ) ∼ = M ⊕ · · · ⊕ M (n-times) as R-modules.
[3 + 3 + 2 = 8].
1
6. Let R be a commutative ring, and let M, N be finite free R-modules.
Prove that HomR (M, N ) is a finite free R-module by producing an explicit
basis. [5]
Additional Exercises.
1. Let R be an integral domain, and M be an R-module. An element
m ∈ M is called a torsion element of M if there is 0 6= r ∈ R such that
rm = 0. Let T (M ) denotes the set of all torsion elements of M . If T (M ) =
{0}, we say that M is torsion-free, and if T (M ) = M , we say that M is a
torsion-module.
(i) Show that T (M ) is a submdoule of M ; and M/T (M ) is torsion-free.
If N is another R-module and φ : M −→ N is an R-module homomorphism,
then φ(T (M )) ⊆ T (N ).
(ii) Consider R2 as R[x]-module via the linear map T : R2 −→ R2 defined
by (u, v) 7→ (v, 0). Find AnnR[x] (R2 ). Show that that R2 is a torsion R[x]-
module.
(iii) If M be a finitely generated and if {x1 , x2 , · · · , xn } is a set of gener-
ators for the module M , then prove that
AnnR (M ) = AnnR (x1 ) ∩ AnnR (x2 ) ∩ · · · ∩ AnnR (xn )
where AnnR (xi ) := {r ∈ R : rxi = 0} is an ideal of R.
2. Let R be a commutative ring. An R-module M is called simple if
M 6= (0) and the only submodules of M are (0) and M . When R = Z, then
simple Z-modules are nothing but simple abelian groups.
(i) Let M, N be simple R-modules. Show that every non-zero R-module
homomorphism φ : M −→ N is an isomorphism.
(ii) An R-module M is simple if and only if M ∼
= R/m for some maximal
ideal m of R.
3. Let R be a ring, and let A, B be two non-empty sets. Let (FA , fA )
and (FB , fB ) be the free R-modules on the sets A and B respectively. Prove
that any bijection φ : A −→ B can be extended to a unique R-module
isomorphism Φ : FA −→ FB , i.e. Φ ◦ fA = fB ◦ φ.
2
4. Give an example to show that submodule of a free-module need not
be free.
5. Let R be a commutating ring, M be a free R-module with a basis
{ai : i ∈ Λ} and let I be an ideal of R. Let π : M −→ M/IM be the natural
R-module homomorphism. Show that M/IM is a free R/I-module with a
basis {π(ai ) : i ∈ Λ}.
6. Let R be a commutative ring, M be an R-module and let M1 , · · · , Mn
∼ n
Sn M = ⊕i=1 Mi . If Ai is a basis of Mi , for each
be submodules of M such that
1 ≤ i ≤ n, then show that i=1 Ai is a basis of M .
7. Let R be a commutative ring, and let M be a free R-module with
a basis {ai : i ∈ I}. Then for every R-module N , there is an R-module
isomorphism
HomR (M, N ) ∼
Y
= Ni
i∈I
where Ni = N, ∀i ∈ I.
