in a principal ideal domain ring let be lcm (a,b) = e , prove that theres a gcd(a,b) that exists in any ring
9873
GCD in a PID persists in extension domains
102757
Prove that either $m$ divides $n$ or $n$ divides $m$ given that $\operatorname{lcm}(m,n) + \operatorname{gcd}(m,n) = m + n$?
83871
GCDs in integral domains are unique up to unit factors (associates)
52483
Showing that $(m)\cap (n)=(\operatorname{lcm}(m,n))$ and $(m)+(n)=(\gcd(m,n))$ for any $m,n\in\mathbb{Z}$
53367
In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?
61383
How do I see that every left ideal of a square matrix ring over a field is principal?
49294
prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
1425
The Chinese Remainder Theorem for Rings.
99198
Principal ideal and free module
50396
How can I show that $ab \sim \gcd (a,b) {\operatorname{lcm} (a,b)}$ for any $a,b \in R \setminus \{0\}$?
74314
Noetherian ring and the same prime divisors
71392
Is $\operatorname{lcm}(a,\gcd(b,c))=\gcd(\operatorname{lcm}(a,b),\operatorname{lcm}(a,c))$
154495
Prove that a nontrivial subring of a ring that is a domain has the same identity
6410
Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
18186
Prove that ideal $I = \langle X^2,X+1\rangle$ is principal or not (in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$ )
48452
Proof that $\gcd$ divides $\operatorname{lcm}$
7090
Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$