Integral Domain

[tegra97]

Hello.
Im having trouble getting started on this problem and am having trouble with rings in general. The problem is

Find all units, zero-divisors, idempotents, and nilpotent elements in Z_3 (+) Z_6. (+) is the external direct product.

The chapter is very vague and doesn't even talk about idempotents and nilpotent. If someone could give me some type of explanation to help get me started would be great. Thanks!

 

[daon]

\(\displaystyle Z_3 X Z_6\) = \(\displaystyle \{\(a,b\); \,\, a \in Z_3 \,\, b \in Z_6\}\)

Note that this is NOT an integral domain. Integral domains have either characteristic 0 or prime. Plus, (2,2)(0,3) = (0,0). It has zero divisors.


Idepotent: \(\displaystyle a^2=a\)

\(\displaystyle (a,b)(a,b)=(a^2,b^2)=(a,b)\) When?


Nilpotent: \(\displaystyle a^n=0\) for some n.

\(\displaystyle (a,b)^n = (a^n,b^n)=0\) When?

 

[tegra97]

Sorry, I used Intergal Domain as the title because the chapter was on Integral domains. How would we find the units? thanks you've been very helpful.

 

[daon]

Units are those elemnts (a,b) in which there exists some element (c,d) so that (a,b)(c,d)=(1,1) (the idenity element for multiplication in your direct sum)..

The idenity is always a unit since (1,1)(1,1)=(1,1). You will need to find two ordered pairs that multiply to (1 mod 3, 1 mod 6). If you do, both of those ordered pairs are units. Gnerally, they come in pairs unless what you find is its own inverse.