Ring of Integers Isomorphism Problem

Click For Summary
SUMMARY

The discussion centers on the isomorphism between the ring of integers modulo N, denoted ZN, and the direct product of the rings of integers modulo A and B, denoted ZA x ZB, where A and B are coprime positive integers. The user proposes a mapping function f(n) = (n mod A, n mod B) and successfully demonstrates that it is homomorphic and injective. The conclusion drawn is that since ZN and ZA x ZB share the same cardinality and f is injective, f must also be surjective, thus establishing the isomorphism. The Chinese Remainder Theorem is referenced as a critical tool in this proof.

PREREQUISITES
  • Understanding of ring theory and isomorphisms
  • Familiarity with the Chinese Remainder Theorem
  • Knowledge of modular arithmetic
  • Basic concepts of homomorphisms and injective functions
NEXT STEPS
  • Study the Chinese Remainder Theorem in detail
  • Explore proofs of isomorphisms in ring theory
  • Practice problems involving homomorphic mappings
  • Investigate applications of modular arithmetic in number theory
USEFUL FOR

Mathematics students, particularly those studying abstract algebra, educators teaching ring theory, and anyone interested in the properties of modular arithmetic and isomorphisms.

e(ho0n3
Messages
1,349
Reaction score
0
Homework Statement
Let N = AB, where A and B are positive integers that are relatively prime. Prove that ZN is isomorphic to ZA x ZB.

The attempt at a solution
I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to conclude, since ZN and ZA x ZB have the same cardinality and f is injective, that f is surjetive? In any case, given an (a, b) in ZA x ZB, I've been trying to find an n such that f(n) = (a, b) without success. Any tips?
 
Physics news on Phys.org
Recall the Chinese remainder theorem.
 
Good tip. Thanks.
 

Similar threads

Showing when quadratic integer rings are isomorphic
  • · Replies 7 ·
Replies
7
Views
2K
Having all subgroups normal is isomorphism invariant
  • · Replies 1 ·
Replies
1
Views
2K
Showing isomorphism between fractions and a quotient ring
  • · Replies 1 ·
Replies
1
Views
2K
##R[x,y]## and ##R[y,x]## are Ring Isomorphic
  • · Replies 3 ·
Replies
3
Views
2K
Showing that GL(F_p) is isomorphic to an automorphism group
  • · Replies 1 ·
Replies
1
Views
2K
Show that Z_12^* and Z_8^* are isomorphic groups
  • · Replies 1 ·
Replies
1
Views
2K
1st Isomorphism thm for dihedral gps
  • · Replies 5 ·
Replies
5
Views
2K
U(n) as an External Direct Product
  • · Replies 11 ·
Replies
11
Views
2K
Showing that rings kZ and lZ, where l≠k are not isomorphic.
  • · Replies 5 ·
Replies
5
Views
3K
Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K