Chào mừng quý vị đến với Website của Phạm Xuân Thịnh.
Problem about Ring and Module
1. ( Chinese remainder theorem)
Let be ideals of the ring
. If
for
, prove:
(a)
(b) The rings and
are isomorphic
2. Let be ideals of the ring
. If
is prime and
, prove that
for some
. In particular if
,then
for some
3. Let be ideais of the ring
. If
and
is a prime ideal for
,prove that
for some
4. Let be a ring and
an ideal. If
is a prime ideal such that
prove that
and show an example where the strict inequality holds
5.Let be an ideal of the ring
. If all the minimal primes of
have the same height and
is an prime ideal such that
prove that
6. Let be an ideal of the ring
. Then a prime ideal
of
is an associated prime of
if and only if
for some
7. Let be an
module and
an ideal of
contained in
. Note that
inherits a structure of
module . Prove that
if and only if
8. Let and
be polynomial rings over field
. If
is an ideal of
and
, then
(a) , where
is a prime ideal of
,and
(b) is an associated prime of
9. Let be a ring and
a proper ideal of
. If every element in
is either nilpotent or invertible , prove that
is a primary ideal
10. If is a polynomial ring over a ring
and
a primary ideal of
,then
is a primary ideal of
Phạm Xuân Thịnh @ 14:10 26/11/2009
Số lượt xem: 677
Exercise 1:
(a) Evidently
.
Coversely,we prove it by induction on
. The case 
,suppose 
,then
Suppose
and result true for 
. Let 
. Since 
we have 
Therefore
. Simple
(b) We have
is surjective with
. Now, the result is simpled from isomorphic theorem
Exercise 2:
Suppose
, then 
such that 
. Since 
is a prime ideal ,therefore 
. But 
. Contradiction !!!
If
,then 