Let R be a PID, B a torsion R module and p a prime in R. Prove that if $pb=0$ for some non zero b in B, then $\text{Ann}(B)$ is a subset of (p)
7067
Torsion Modules Annihilators
141802
Prove that free modules are projective
135586
Prove that the preimage of a prime ideal is also prime.
45219
Proving that $C$ is a subset of $f^{-1}[f(C)]$
19822
For $D$ a GCD domain , let $a,b,x \in D \setminus \{0\}$ , then is it true that $\gcd (ax,bx)=x \cdot \gcd (a,b)$?
83050
Show that $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.
19838
Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R
145752
Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$
154305
Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
83009
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
23153
$\bmod p\!:\ x^2\equiv 1\iff x\equiv \pm1\ $ for prime $p$
344
Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
4295
Let $N$, $K$ be sub-modules of $M$ with $I=\mathrm{Ann}(N)$, $J=\mathrm{Ann}(K)$. Show $I+J$ is a proper subset of $\mathrm{Ann}(N \cap K)$.
105313
Existence of minimal prime ideal contained in given prime ideal and containing a given subset
131971
If $k>0$ is a positive integer and $p$ is any prime, when is $\mathbb Z_p[\sqrt{k}] =\{a + b\sqrt k~|~a,b \in\mathbb Z_p\}$ a field.
83721
If a prime $p\mid ab$, then $p\mid a$ or $p\mid b\ $ [Euclid's Lemma]
150986
For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$