Direct Sum of Rings?

  • Thread starter altcmdesc
  • Start date
  • #1
66
0
What's the difference (if any) between a direct sum and a direct product of rings?

For example, in Ireland and Rosen's number theory text, they mention that, in the context of rings, the Chinese Remainder Theorem implies that [tex]\mathbb{Z}/(m_1 \cdots m_n)\mathbb{Z}\cong\mathbb{Z}/m_1\mathbb{Z}\oplus\cdots\oplus\mathbb{Z}/m_n\mathbb{Z}[/tex]. The way I've been taught is that every [tex]\oplus[/tex] should be replaced with a [tex]\times[/tex] so that we are talking about direct products, not direct sums.
 

Answers and Replies

  • #2
quasar987
Science Advisor
Homework Helper
Gold Member
4,783
18
Yes, direct sum and direct product coincide for finitely many summands. You can check out the definition of both product on wikipedia.
 
  • #3
Hurkyl
Staff Emeritus
Science Advisor
Gold Member
14,916
19
He's not dealing with modules -- he's dealing with rings.
 

Related Threads on Direct Sum of Rings?

  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
5
Views
5K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
6
Views
3K
Replies
5
Views
4K
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
4K
Top