Prove Isomorphism: R x S & S x R

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

The discussion centers on proving the isomorphism between the rings R x S and S x R for any rings R and S. Participants emphasize the necessity of establishing a bijection that preserves both addition and multiplication. The suggested approach involves utilizing a "natural" mapping, where an element (r1, s1) from R x S is mapped to (s1, r1) in S x R. This mapping effectively demonstrates the isomorphic relationship between the two Cartesian products.

PREREQUISITES
  • Understanding of ring theory and properties of rings
  • Familiarity with Cartesian products in set theory
  • Knowledge of bijections and their role in establishing isomorphisms
  • Basic concepts of addition and multiplication in algebraic structures
NEXT STEPS
  • Study the properties of isomorphic rings in abstract algebra
  • Learn about bijective functions and their significance in mathematics
  • Explore examples of natural mappings in algebraic structures
  • Investigate the implications of ring isomorphisms in advanced mathematics
USEFUL FOR

Students of abstract algebra, mathematicians focusing on ring theory, and educators seeking to explain the concept of isomorphism in algebraic structures.

stripes
Messages
262
Reaction score
0

Homework Statement



Show that for any rings R and S, R x S and S x R are isomorphic, and A x B is the cartesian product, or ordered pairs. So an element of R x S can be written as (r1, s1).

Homework Equations





The Attempt at a Solution



So I have to show that there is a bijection from R x S to S x R, and this bijection must preserve addition and multiplication. This is tough for me since the mapping from R x S to S x R could be anything! How can I even start if I don't have this function or mapping?
 
Physics news on Phys.org
hi stripes! :smile:
stripes said:
… the mapping from R x S to S x R could be anything!

yes it could

but you're in charge, and you can choose any mapping you like :smile:

go for the "natural" mapping …

what do you think it would be really neat for (r1, s1) to be mapped onto? :wink:​
 

Similar threads

Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
  • · Replies 1 ·
Replies
1
Views
1K
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
  • · Replies 9 ·
Replies
9
Views
2K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
Showing when quadratic integer rings are isomorphic
  • · Replies 7 ·
Replies
7
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##a + b = b + a## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
  • · Replies 2 ·
Replies
2
Views
1K
Showing isomorphism between fractions and a quotient ring
  • · Replies 1 ·
Replies
1
Views
2K
1st Isomorphism thm for dihedral gps
  • · Replies 5 ·
Replies
5
Views
2K