# Examples of ring where doesn't exist lcm and gcd

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.

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.