My question is:
"Examples of ring that doesn't exist lcm and gcd of any elements"
The ring preferely has to be commutative and unitary or olny unitary( as matrix ring). I woul use these two theorems:
Theorem 1
Let $a_1,a_2,a_3,……,a_n$ be nonzero elements of a ring $R$. Then $a_1,a_2,a_3,……,a_n$ have a least common multiple if and only if the ideal $\cap (a_i)$ is principal.
Theorem 2
Let $a_1,a_2,a_3,……,a_n$ be nonzero elements of the ring $R$. Then $a_1,a_2,a_3,……,a_n$ have a greatest common divisor $d$, expressible in the form
$$d=r_1a_1+r_2a_2+…+r_na_n \quad (r_i \in R)$$
if and only if the ideal $(a_1……a_n)$ is principal.
Best Answer
I found on the Web and, also on mathstack, several examples and I decided to write an answer.
Example of ring where the gcd of two elements doesn't exist:
Consider the ring $\Bbb Z[\sqrt{-d}]=\{a+bi\sqrt{-d} : a,b\in \Bbb Z\}$, $d\ge 3$ ($d$ free-square). Then in this paper D. Khurana has proved that:
Let $a$ be any rational integer such that $a\equiv d\quad (mod\quad 2)$ and let $a^2 + d = 2q$. Then the elements $2q$ and $(a+ i\sqrt{d})q$ do not have a $GCD$.
Two examples of ring where the lcm of two elements doesn’t exist:
I use the following theorem: Let $D$ be a domain and $a,b\in D$. Then, $\text{lcm}(a,b)$ exists if and only if for all $r\in D\setminus\{0\}$, $\gcd(ra,rb)$ exists.
$1)$Consider the ring $\Bbb Z[\sqrt{-d}]=\{a+bi\sqrt{-d} : a,b\in \Bbb Z\}$, $d\ge 3$ ($d$ free-square); and a rational integer $a$ such that $a\equiv d$ $(mod\quad 2)$, then $lcm(2,a+i\sqrt{d})$ doesn't exist. Indeed this follows from the previous theorem. Note that $GCD(2,a+i\sqrt{d})=1$.
$2)$ let $R$ be the subring of $\Bbb Z[x]$ consisting of the polynomials $\sum_i c_ix^i$ such that $c1$ is an even number. If we consider $p_1(x)=2$ and $p_2(x)=2x$, $lcm(p_1, p_2)$ doesn't exist.