$R$ 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?
131497
Maximal ideal not containing the set of powers of an element is prime
90314
Maximal ideals in $R[x]$
108896
Prime ideal in a polynomial ring over an integrally closed domain
87382
Non-invertible elements form an ideal
76813
The ring $ℤ/nℤ$ is a field if and only if $n$ is prime
138826
To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.
5360
Ideal contained in a finite union of prime ideals
43230
Is $\langle1-i\rangle$ a maximal ideal in $\mathbb{Z}[i]$?
80232
Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?
15904
Prime ideal and nilpotent elements
104486
Ideal in $\mathbb Z[x]$ which is not two-generated
148788
Polynomial ring ideal. Show that $(f) \cap R_N = (0) \iff f \notin R_N$
53367
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
138130
If $I$ is a maximal ideal of $R$, why is $R/I$ a field?
9792
Is there an example of a power of prime ideal in a polynomial ring that is not primary?
135682
Are distinct prime ideals in a ring always coprime? If not, then when are they?
113582
How can I find an element $x\not\in\mathfrak mM_{\mathfrak m}$ for every maximal ideal $\mathfrak m$