Homomorphism of Rings & Is Map f:C->Z a Homo?

  • Context: MHB 
  • Thread starter Thread starter Stephen88
  • Start date Start date
  • Tags Tags
    Rings
Click For Summary
SUMMARY

The discussion confirms that the map f: C -> Z defined by f(a + bi) = a is not a homomorphism of rings. The reasoning provided shows that while f(x + y) = f(x) + f(y) holds true, f(xy) does not equal f(x)f(y), specifically f((ac - db) + (ad + bc)i) ≠ ac - db. Therefore, the conclusion that the map is not a homomorphism is established as correct.

PREREQUISITES
  • Understanding of ring theory and homomorphisms
  • Familiarity with complex numbers and their representation
  • Basic knowledge of algebraic operations on rings
  • Concept of mappings between mathematical structures
NEXT STEPS
  • Study the properties of ring homomorphisms in abstract algebra
  • Explore examples of valid homomorphisms between different algebraic structures
  • Learn about the implications of mappings in complex analysis
  • Investigate the role of complex numbers in ring theory
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone studying the properties of ring homomorphisms and complex number mappings.

Stephen88
Messages
60
Reaction score
0
I'm trying to see if the map f:C->Z,f(a+bi)=a is a homomorphism of rings.
Let x=a+bi and y=c+di...then f(x+y)=a+c=f(x)+f(y) but f(xy)=f((ac-db)+(ad+bc)i)=ac-db/= f(x)f(y)...so the map is not a homomorphism of rings.
Is this correct?
 
Physics news on Phys.org
StefanM said:
Is this correct?
Yes. (Smile)
 

Similar threads

Undergrad Correct constructed example of Abelian Monoid?
  • · Replies 0 ·
Replies
0
Views
928
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
980
Undergrad Meaning of "identify" & variations in different math context
  • · Replies 5 ·
Replies
5
Views
1K
Undergrad Localizing single variable quotient polynomial ring at a prime ideal
  • · Replies 6 ·
Replies
6
Views
1K
Undergrad Trouble understanding an online solution to an exercise in Dummit & Foote
  • · Replies 21 ·
Replies
21
Views
1K
Undergrad Basic Question about a Ring Homomorphisms
  • · Replies 10 ·
Replies
10
Views
3K
MHB Basic Question on Ring Homomorphisms
  • · Replies 1 ·
Replies
1
Views
2K
MHB Second Isomorphism Theorem for Rings .... Bland Theorem 3.3.15 .... ....
  • · Replies 6 ·
Replies
6
Views
2K
MHB A question about surjective module-homomorphisms
  • · Replies 3 ·
Replies
3
Views
2K
MHB Ring Homomorphisms from Z to Z .... Lovett, Ex. 1, Section 5.4 .... ....
  • · Replies 2 ·
Replies
2
Views
2K