The units of a cartesian product?

chuy52506
Messages
77
Reaction score
0
Find all units in Z12 X Z6 and their inverses.

What i did was find the units of Z12 which are 1,11,5,7 then the ones of Z6 which are 1,5 and take the cartesian product of those two sets?
 
Physics news on Phys.org
Yes, that would be correct!
 
So there would be eight?
1,1 1,5 11,1 11,5 5,1 5,5 7,1 and 7,5?
 
Instead of asking this, why don't you try to prove that the units of the product are precisely the pairs of units?
 
Because it helps to do an example before. So is this correct?
 
micromass already answered that question. If you're still not entirely sure, my advice stands: prove it! (it's quite easy; it's a matter of writing out the definitions)
 

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 Terminologies used to describe tensor product of vector spaces
Replies
7
Views
3K
I Question Regarding Definition of Tensor Algebra
Replies
10
Views
3K
I Dot product in Euclidean Space
Replies
16
Views
2K
Finding Elements of Order 6 in Aut(Z720)
Replies
7
Views
7K
Unit vector in cylindrical coordinates
Replies
1
Views
3K
Kinematics problem involving multiple frames
Replies
6
Views
747
I Dot product of two vector operators in unusual coordinates
Replies
14
Views
2K
I Supremum of a set, relations and order
Replies
35
Views
4K
B Cartesian Space vs. Euclidean Space
Replies
10
Views
2K
B Confused about dot product multiplication
Replies
33
Views
4K

Hot Threads

Recent Insights

Back
Top