Show that if $\mathfrak{p} \subset R$ is a prime ideal, then $S = R \setminus \mathfrak{p}$ is multiplicatively closed.
100553
Complement of a prime ideal is a subset closed under multiplication, containing 1
101637
Let $R$ be an integral domain. If $x \in R$ is prime, then $x$ is irreducible.
138114
Proving the quotient of a principal ideal domain by a prime ideal is again a principal ideal domain
78455
Prove that a nonzero integer p is prime if and only if the ideal (p) is maximal in Z.
19230
Prime/Maximal Ideals in $\mathbb{Z}[\sqrt d]$
56800
Intersection of prime ideals
135586
Prove that the preimage of a prime ideal is also prime.
149203
Symbolic power of a prime ideal is primary
111114
Prove that an ideal in $k[x,y]$ is a prime ideal
105313
Existence of minimal prime ideal contained in given prime ideal and containing a given subset
103518
If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents
53770
Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.
146304
prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero
22716
$A$ is integrally closed if and only if $A_P$ is integrally closed for all maximal ideals $P$ of $A$.
121890
Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
83009
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
145594
Splitting of prime ideals in algebraic extensions