Prove that if $R$ is a principal ideal ring and $S$ is a multiplicatively closed subset of $R$ then $S^{-1}R$ is also a principal ideal ring.
24931
Is the localization of a PID a PID?
131671
Let R be a ring such that $a^2 = a$ , $\forall a$ $\in R$ . Prove that R is commutative.
67203
Complement of maximal multiplicative set is a prime ideal
20877
The set of all nilpotent elements is an ideal
58640
$R$ is an algebra over an infinite field. If $\exists$ ideals s.t. $J\subseteq \bigcup_{k=1}^nI_k$ then $J\subseteq I_k$ for some $k$
42170
$R$ integral domain where every prime ideal is principal then set of non-principal ideals has maximal element
53367
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
90314
Maximal ideals in $R[x]$
16602
Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings.
99567
Proving that an inverse ring homomorphism of an ideal is an ideal?
146801
Primary decomposition of an ideal
129302
Proving whether ideals are prime in $\mathbb{Z}[\sqrt{-5}]$
99198
Principal ideal and free module
81982
Is $\mathbb{Z}[x]$ a principal ideal domain?
84663
$A$ local Noetherian ring with principal maximal ideal implies PIR?
138752
Prove the Radical of an Ideal is an Ideal
40705
If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module