$R$ be a ring and $a\in R$ be a non nilpotent element and define $S=\{a,a^2,a^3,\cdots\}$ and if $P$ is maximal ideal of $F$, is $P$ prime ideal?

Asked Jul 01 '20 at 08:02

Active Jul 01 '20 at 12:07

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$R$ be a ring and $a\in R$ be a non nilpotent element and define $S=\{a,a^2,a^3,\cdots\}$ and $F=\{A\subseteq R| A-ideal \;, S\cap A=\emptyset\}$

If $P$ is maximal ideal of $F$, is $P$ prime ideal?

I am self studying abstact algebra but I didnot understand first of all is the set $F$ a ring? In what operations? Furthermore, how one can show this ?

asked Jul 01 '20 at 08:02

Jale'de jaled

What is the link between $P$ and the other sets here ? – FiMePr Jul 01 '20 at 08:30
3

$F$ is a set of ideals and as such it is a partially ordered set (under inclusion). The correct phrase would be $P$ is a maximal element of $F$. In this case one can show that $P$ is a prime ideal. – Kapil Jul 01 '20 at 08:32
@FiMePr I recheck, every set is linked, I dont see your point? – Jale'de jaled Jul 01 '20 at 08:33
@Kapil Thanks ! – FiMePr Jul 01 '20 at 08:35
I think F is not a ring,,,for suppose take R= Z( set of all integers ) as ring,,,S={1},,,then clearly 2Z and 3Z doesn't contain 1,,,,but their addition takes 1,,, – A learner Jul 01 '20 at 08:35
For the proof,,,,you will have to add some more information. – A learner Jul 01 '20 at 08:40
@Subhajit but your example is supporting the argument, can you give me counter example? P is maximal but is not prime? – Jale'de jaled Jul 01 '20 at 12:02

$R$ be a ring and $a\in R$ be a non nilpotent element and define $S=\{a,a^2,a^3,\cdots\}$ and if $P$ is maximal ideal of $F$, is $P$ prime ideal?

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