Zero product property for integers

de1irious
Messages
20
Reaction score
0
Hi, I am just wondering if the zero product property (ab=0 implies a=0 or b=0) can be proven on the integers, or is it directly axiomatic to the defining of the integers? Also, where might I find a definition of the integer axioms? Thanks a lot,
 
Physics news on Phys.org
Last edited by a moderator:

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

A Near-Rings with Noncommutative Addition and Two-Sided Distributivity
Replies
4
Views
2K
I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?
Replies
48
Views
3K
I Base Pi Integers: Isomorphism with Rationals?
Replies
33
Views
4K
Why this property of the product of two matrices
Replies
5
Views
2K
I Localising a non integral domain at a prime
Replies
17
Views
1K
Set properties of even (##E##) and odd (##I##) integers
Replies
5
Views
1K
I Understanding zero divisors & ##\mathbb{Z_m}## in Ring Theory
2
Replies
55
Views
6K
Query about how the domain of a binomial coefficient was calculated
Replies
2
Views
2K
Prove the set of integers is a commutative ring with identity
Replies
2
Views
4K
I Dfns group action & group, written in terms of the other
Replies
26
Views
379

Hot Threads

Recent Insights

Back
Top