Factor Ring of a Ring: Example of Integral Domain with Divisors of 0

  • Thread starter Thread starter tinynerdi
  • Start date Start date
  • Tags Tags
    Ring
Click For Summary

Homework Help Overview

The discussion revolves around the concept of factor rings in the context of ring theory, specifically focusing on examples that illustrate the relationship between rings with zero divisors and integral domains.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore examples of factor rings, initially suggesting ZxZ/5Z and later reconsidering the relationship between Z and ZxZ.

Discussion Status

The conversation reflects a mix of attempts to clarify the definitions and relationships between the rings discussed. Some participants question the validity of the initial examples and definitions, indicating a productive exploration of the topic.

Contextual Notes

There is confusion regarding the structure of ZxZ as a subring and its relation to Z, highlighting potential misunderstandings about ring properties and definitions.

tinynerdi
Messages
9
Reaction score
0

Homework Statement


give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain.


Homework Equations


since we know that ZxZ is a zero divisor and 5Z is an integral domain.


The Attempt at a Solution


So, ZxZ/5Z =~(isomorphic to) Z/5Z=~ Z_5.
 
Physics news on Phys.org
ZxZ/5Z doesn't even make much sense. 5Z isn't a subring of ZxZ.
 
I forgot about that.. let try Z/ZxZ =~ Z Since ZxZ is a subring of Z.
 
tinynerdi said:
I forgot about that.. let try Z/ZxZ =~ Z Since ZxZ is a subring of Z.

How are you considering ZxZ as a subring of Z? ZxZ is ordered pairs of integers, isn't it? Z is just integers.
 

Similar threads

  • · Replies 3 ·
Replies
3
Views
3K
  • · Replies 13 ·
Replies
13
Views
3K
  • · Replies 1 ·
Replies
1
Views
2K
  • · Replies 10 ·
Replies
10
Views
2K
  • · Replies 4 ·
Replies
4
Views
3K
Replies
7
Views
4K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 55 ·
2
Replies
55
Views
8K
Replies
2
Views
3K
  • · Replies 24 ·
Replies
24
Views
3K