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3
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2answers
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The greatest common divisor of $a$ and $b$ is a linear combination of $a$ and $b$. In general, in what kind of ring does this hold?
In $\mathbb{Z}$, the greatest common divisor of $a$ and $b$ is a linear combination of $a$ and $b$.
This generalizes to Euclidean domains since Euclid's algorithm works. Moreover this statement ...
0
votes
6answers
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Proof that $\mathbb{Z}$ has no zero divisors
Everyone knows the rules of zero divisors like
$$\forall \alpha,\beta\in\mathbb{R}\;:\;\alpha\cdot\beta = 0\Rightarrow\alpha=0\vee \beta=0.$$
But how can I prove it for $\mathbb{Z}$? My first try was ...
5
votes
3answers
351 views
If $R$ is a commutative ring with identity, and $a, b$ are its elements that are divisible by each other, is it true that they must be associates?
Thank you very much!
My problem is:
If $R$ is a commutative ring with identity, and $a, b$ are its elements that are divisible by each other, is it true that they must be associates?
Here, $a$ ...
