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In lecture I encountered the following remark:

In $\mathbb{Z}[X^2,X^3]$ the $lcm(X^2,X^3)$ and $gcd(X^2 \cdot X^3, X^3 \cdot X^3)$ do not exist.

I do not understand this at all, for as far as I can see we can "normally" multiply the $X$'s with each other, so intuitively I would say that $X^3 = lcm(X^2,X^3)$ and $X^4 = gcd(X^2 \cdot X^3, X^2 \cdot X^2)$; since $X^2 \cdot X^3 = X^5$.

Could you please explain that to me?

3nondatur
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1 Answers1

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The GCD is the divisor which all other common divisors divide. $X^3$ and $X^4$ are both common divisors, but neither divides the other in this ring because $X$ is not a part of it.

Similarly, the LCM is the common multiple which divides all other common multiples. $X^5$ and $X^6$ are common multiples of $X^2$ and $X^3$, but neither divides the other.

eyeballfrog
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