Abstract Algebra: Integral Domain

[late halo]

I'm guessing you're thinking of only real numbers and themselves right?

Try using matricies and see if you can find two matrices that multiply to equal a zero matrix

[tight tinsel]

There are actually infinitely many rings that aren’t integral domains. Consider any non-zero $n$ and the factor ring $\mathbb{Z}/n^2\mathbb{Z}$. Can you see which element is a zero divisor in such a ring?

[silk dove]

Ellie <3

[half inlet]

Or take the integers mod 6

2*3=0

[loud nexus]

That's not quite right, 9 is a square but the integers mod 3 form a field, in particular an integral domain.

What you probably mean to do is consider rings of the form $\mathbb{Z} / n^2 \mathbb{Z}$, in which $n$ is a zero divisor.

[silk dove]

redsquirrel

[tight tinsel]

Whoops yes

You can also just pick any non-prime positive integer

[steel crow]

Consider $\begin{bmatrix}
1 & 0 \ 0 & 0
\end{bmatrix}
\begin{bmatrix}
0 & 0 \ 0 & 1
\end{bmatrix}$

[silk dove]

HChan

[steel crow]

you could think of it that way, I guess

[steel crow]

no not really

just clever position of the zeros

if a matrix is filled with mostly zeros chances are it is a zero divisor

[late halo]

Well, clearly this is a non zero element multiplied by another non zero element that still gets a zero element

well it's just an example of your idea right?

The real number system with multiplication can't do this clearly but that's not very general

well often times we are given a certain system and asked if these properties apply or not

objects as in...

not all matrices multiply with eachother to get zeros clearly

but we can figure out at least one does

[late halo]

Good to hear!!

[tight tinsel]

All objects having a certain property can be characterized by that property itself. But if you want a feel for what something is, it’s good to think of examples

For integral domains, you can look at any field (b/c all fields are integral domains) for an example of an integral domain