So I have a statement that say "Let D be a principal ideal domain, if a is unit then (a)=D". How it could be? I don't understand, can anyone explains it to me? Thank you.
Asked
Active
Viewed 228 times
0
-
That is because as $a$ is a unit the ideal generated by $a$ contains $1$, hence all multiples of $1$. It is actually true for any unital ring. – Bernard Mar 24 '21 at 23:42
-
1Because for any given $x$, $x=xa^{-1}\cdot a\in (a)$ – Mar 24 '21 at 23:42
-
This is true for any ring, not just PID's. An ideal $I$ is the entire ring if and only if $I$ contains a unit. – JLinsta Mar 24 '21 at 23:48
-
Okay, I understand all of your explanation. Thank you guys. – Dicky Ardiyantoro Mar 25 '21 at 00:02