Problem Set 2
1) Let n

, n

1 and let R be the ring of n n matrices with entries from a field F . Let M be the set of n×n matrices with arbitrary elements of
F in the first column and zeros elsewhere.
Show that M is a submodule of R when R is considered as a left modulre over itself, but M is not a submodule of R when R is considered as a right R -module.
2) Assume that rm

0 for some r

R and some m M with m

0 . Prove that r does not have a left inverse (i.e., there is no s

R such that sr

1 ).
3) a) If N is a submodule of M , the annihilator of N in R is defined to be: { r

R for all n

N }. Prove that the annihilator of N in R is a two sided ideal of R .
| rn

0
b) If I is a right ideal of R , the annhilator of I in M defined to be { m M | a

I }. Prove that the annihilator of I in M is a submodule of M . am

0 for all
4) If M is a finite abelian group then M is naturally a ℤ - module. Can this action be extended to make M into a ℚ -module? Why or why not?
5) Let K ⊕ K′ and L ⊕ L′ be direct sums of submodules of M such that K ⊕ K′=L ⊕ L′. Show that K=L does not necessarily imply K′=L′.
6) a) An R-module is called irreducible if M≠0 and if M and 0 are the only submodules of M.
Show that M is irreducible if and only if M≠0 and M is a cyclic module with any nonzero element as generator. Determine all the irreducible ℤ -modules.
b) Assume R is commutative. Show that an R-module M is irreducible if and only if M is isomorphic (as an R-module) to R ╱ I, where I is a maximal ideal of R.

c) Show that if M ₁ and M ₂ are irreducible R-modules, that any nonzero R-module homomophism from M ₁ to M ₂ is an isomorphism. Deduce that if M is irreducible then
End ( M )
R
is a division ring (Schur's Lemma).
7) An element e

R is called a central idempotent if e
2  e central idempotent in R , prove that M
 eM (1 ) .
and er
 re for all r

R . If e is a
8) Let N
1
, , N be a family of R -submodules of an R -module M .Suppose M= k
N i

( N
1
 
N i

1

N i

1
 
N k
) .
Show that M k i

1
N i
M
N
1
  M
N k
. (analogue of the Chinese Remainder Theorem).
9) Show that every ideal of ℤ is free as a ℤ -module.
10) Let R be a commutative ring and let , , isomorphisms of R-modules:
a)
be R
R
(

, )
R
( , )

R
( , ) .
-modules. Prove the following
b)
R
( ,

)
R
( , )

R
( , ) .