MHB Solving Nilpotent Problem in Abstract Algebra

  • Thread starter Thread starter cbarker1
  • Start date Start date
cbarker1
Gold Member
MHB
Messages
345
Reaction score
23
Dear Everyone,

I am stuck with an exercise problem. The problem states from Dummit and Foote Ed. 2 Abstract Algebra: "An element $x$ in $R$ (where $R$ is a ring with 1) is called nilpotent if $x^{m}=0$ for some $m \in \Bbb{Z}^{+}$. Show that if $n=a^{k}b$ for some $a,b \in \Bbb{Z}$, then $\overline{ab}$ is a element of $\Bbb{Z}/n\Bbb{Z}$."

My attempt:

Example: When $n=6=3\cdot 2$, then the only element will be $\overline{3*2}$ in $\Bbb{Z}/6\Bbb{Z}$.

Proof: Suppose $n=a^{k}b$ for some $a,b \in \Bbb{Z}$. (Do I need some cases? If so, three cases?)

Thanks,
Cbarker1
 
Last edited:
Physics news on Phys.org
Cbarker1 said:
"An element $x$ in $R$ (where $R$ is a ring with 1) is called nilpotent if $x^{m}=0$ for some $m \in \Bbb{Z}^{+}$. Show that if $n=a^{k}b$ for some $a,b \in \Bbb{Z}$, then $\overline{ab}$ is a element of $\Bbb{Z}/n\Bbb{Z}$."
Could you define the notation $\overline{ab}$? Also, it is strange that the statement one is asked to prove does not mention the concept "nilpotent", which is defined right before that.
 
$\overline{ab}:=a * b $(mod n)

Sorry. I typed quickly and I forgot about the most important information. If $n=a^kb$ for some $a,b\in \Bbb{Z}$, then $\overline{ab}$ is a nilpotent element of $\Bbb{Z}/n\Bbb{Z}$. Again, I am sorry. Cbarker1
 
Last edited:
Consider $\overline{ab}^{\,k}$.
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'
This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
Back
Top