Subring of Z₂₈ & Isomorphism: S={0,4,8,12,16,24}

  • Thread starter Thread starter dash
  • Start date Start date
  • Tags Tags
    Isomorphism
Click For Summary
SUMMARY

The set S = {0, 4, 8, 12, 16, 24} is confirmed as a subring of Z₂₈ under the usual definition of a ring, which includes the requirement for a multiplicative identity. The mapping Ø: Z₇ → S defined by Ø(x) = 8x mod 28 is proven to be an isomorphism, establishing a one-to-one correspondence between Z₇ and the subring S. The discussion clarifies that while S meets the criteria for a ring, it does not qualify as a subring of Z₂₈ due to the absence of a multiplicative identity.

PREREQUISITES
  • Understanding of ring theory and its definitions, particularly the concept of a multiplicative identity.
  • Familiarity with modular arithmetic, specifically Z₂₈ and Z₇.
  • Knowledge of isomorphisms in algebraic structures.
  • Basic skills in mathematical proof techniques.
NEXT STEPS
  • Study the properties of subrings in ring theory.
  • Explore the concept of isomorphisms in greater depth, particularly in modular arithmetic.
  • Learn about the implications of multiplicative identities in algebraic structures.
  • Investigate examples of other subrings within different modular systems.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in the properties of rings and isomorphisms in modular arithmetic.

dash
Messages
7
Reaction score
0
Show that the set S = { 0, 4, 8, 12, 16, 24} is a subring of Z subscript 28. Then prove that the map Ø: Z subscript 7 → S given by Ø(x) = 8x mod 28 is an isomorphism
 
Physics news on Phys.org
You need to show what you've tried before getting help.
 
dash: The usual definition of the word "ring" requires it to have a multiplicative identity. Are you using the usual definition? Or are you using an alternate definition that doesn't make such a requirement?

It doesn't really matter for what you're actually trying to do -- but if you are using the usual definition of ring, then the subset {0, 4, 8, 12, 16, 24} of Z / 28 is a ring, but not a subring of Z / 28.
 

Similar threads

Showing when quadratic integer rings are isomorphic
  • · Replies 7 ·
Replies
7
Views
2K
Show that Z_12^* and Z_8^* are isomorphic groups
  • · Replies 1 ·
Replies
1
Views
2K
Rings Isomorphism: Proving R & R_2 Subrings of Z & M_2(Z)
  • · Replies 8 ·
Replies
8
Views
2K
Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
  • · Replies 2 ·
Replies
2
Views
2K
Is Z7[x] Isomorphic to Z?
  • · Replies 1 ·
Replies
1
Views
2K
[True or False] Subring of R isomorphic to R/I?
  • · Replies 12 ·
Replies
12
Views
3K
Testing whether a binary structure is a group
  • · Replies 1 ·
Replies
1
Views
1K
Showing that a subgroup of Sym(4) is isomorphic to D_8
  • · Replies 1 ·
Replies
1
Views
2K
How should I find ## x_{2}(t) ## for this nonlinear integro-differential equation?
  • · Replies 6 ·
Replies
6
Views
3K
U(n) as an External Direct Product
  • · Replies 11 ·
Replies
11
Views
2K