Let [tex]R[/tex] be a ring with [tex]1_R[/tex]. If [tex]M[/tex] is an

  • Thread starter Thread starter math_grl
  • Start date Start date
  • Tags Tags
    Ring
math_grl
Messages
46
Reaction score
0
Let R be a ring with 1_R. If M is an R-module that is NOT unitary then for some m \in M, Rm = 0.

I'm pretty sure Rm = \{ r \cdot m \mid r \in R \}. While M being not unitary means that 1_R \cdot x \neq x for some x \in M. I'm thinking this problem should be an obvious and direct proof but I can't see it.
 
Physics news on Phys.org


If m ≠ 1m, then consider m - 1m ≠ 0.
 


:blushing:

that's embarassing.
 

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

I Clarifications needed for an exercise in Hungerford's abstract algebra
Replies
23
Views
453
I Meaning of "identify" & variations in different math context
Replies
5
Views
510
MHB Are Hom_R(R, M) and M Isomorphic?
Replies
3
Views
2K
I Definition of minimal ideal using symbolic logic notation
Replies
3
Views
565
I Trouble with a passage in Zariski & Samuel's Commutative algebra text
Replies
5
Views
632
MHB Proving $\text{GL}_{2}(R): Showing Homomorphism & Isomorphism
Replies
1
Views
2K
MHB Why is Sharp's Lemma 1.10 crucial for understanding R-algebras?
Replies
1
Views
2K
MHB Zero divisor for polynomial rings
Replies
1
Views
2K
Demonstrating that a mapping is injective
Replies
1
Views
2K
MHB Demonstrating that a mapping is injective
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top