Module action sends zero to zero?

  • Context: Graduate 
  • Thread starter Thread starter geor
  • Start date Start date
  • Tags Tags
    module Zero
Click For Summary
SUMMARY

In the discussion, the user seeks to prove that for any ring R and R-module M, the product of any element r in R with the zero element of M results in zero (r0 = 0). The proof is established by demonstrating that r0 can be expressed as r(0 + 0), which simplifies to r0 + r0. By canceling r0 from both sides, it is confirmed that 0 = r0, thus validating the initial claim. This fundamental property of modules is crucial for understanding module theory in abstract algebra.

PREREQUISITES
  • Understanding of ring theory and R-modules
  • Familiarity with abelian groups
  • Basic knowledge of algebraic structures and operations
  • Proficiency in mathematical proofs and cancellation laws
NEXT STEPS
  • Study the properties of R-modules in detail
  • Explore the implications of zero elements in algebraic structures
  • Learn about homomorphisms between modules
  • Investigate the relationship between rings and their modules
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in the foundational concepts of module theory and ring theory.

geor
Messages
35
Reaction score
0
Hello,

Excuse me if I ask something obvious here, but I can't
prove it..

Say R is a ring and M an R-module.
So, 0 \in M as M is an abelian group.

I guess r0=0, no?!

Can you tell me how we prove that?

Again, sorry if I ask something obvious..
 
Physics news on Phys.org
It's true:

r0 = r(0 + 0) = r0 + r0. Canceling r0 from both sides gives 0 = r0.
 
adriank said:
It's true:

r0 = r(0 + 0) = r0 + r0. Canceling r0 from both sides gives 0 = r0.

Oh, yes.. :shy:

Thanks a lot!
 

Similar threads

Undergrad Definition of minimal ideal using symbolic logic notation
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Near-Rings with Noncommutative Addition and Two-Sided Distributivity
  • · Replies 4 ·
Replies
4
Views
3K
MHB Basic Question on Modules - Dummit and Foote Chapter 10
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Meaning of "passage to the quotient"?
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Understanding zero divisors & ##\mathbb{Z_m}## in Ring Theory
  • · Replies 55 ·
2
Replies
55
Views
8K
Graduate Proving the Dual of Schanuel's Lemma
  • · Replies 7 ·
Replies
7
Views
2K
Graduate On a finitely generated submodule of a direct sum of modules....
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Trouble with a passage in Zariski & Samuel's Commutative algebra text
  • · Replies 5 ·
Replies
5
Views
2K
Graduate A decreasing sequence of images of an endomorphisme
  • · Replies 18 ·
Replies
18
Views
2K
Undergrad Differences between left vs right actions in some group theory questions
  • · Replies 1 ·
Replies
1
Views
1K