Product Ideal

  • Thread starter Ratpigeon
  • Start date
  • #1
56
0

Homework Statement


Show that if I and J are ideals of the (commutative) ring; R then
S={xy|x in I y in J} is not necessarily an ideal but the set of finite sums
IJ={Ʃ(xvyv)|xvin I yvin J}
is (and called the product ideal).

Homework Equations


An ideal satisfies the properties
For all x, x' in I, all r in R
(i) x+x' in I
(ii) rx in I


The Attempt at a Solution


This seems wrong to me since if I is generated by i and J is generated by j
xy+x'y'= ai*bj+a'i*b'j=(ab+a'b')(ij)=(ab+a'b')i*j which is in S since (ab+a'b')i is in I and j is in J...
and obviously by the same rule;
r(xy) is in S.
Therefore, it seems to me that S and IJ are equivalent sets (since the elements of S could be split up into a sum of product elements) and I don't see how S could _not_ be an ideal.

I haven't been able to find much information about product ideals; but this is a problem in my textbook - Algebra by Michael Artin (second edition) so I'm disinclined to think the lecturer phrased the question wrong...
Please help?
 

Answers and Replies

  • #2
210
0
Attached
 

Attachments

  • 001.jpg
    001.jpg
    16 KB · Views: 336

Related Threads on Product Ideal

Replies
7
Views
1K
  • Last Post
Replies
1
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
0
Views
1K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
9
Views
1K
  • Last Post
Replies
1
Views
681
  • Last Post
Replies
11
Views
5K
Top