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In number theory, for any given integers $a,b$, $a|b$ is defined to mean $a \text{ divides } b$.

What about polynomials division? Or any ring? Is $f(X)|g(X)$ generally used to mean $f(X) \text{ divides } g(X)$?

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Yes, $f(X)\mid g(X)$ or $f\mid g$. –  Frank Science Feb 8 '13 at 5:57
    
Just, I would call it divisibility. I believe division is usually reserved for Euclidean division, when it can be done. –  Andreas Caranti Feb 8 '13 at 6:25

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Yes, that makes perfect sense.

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We even use this notion with the same meaning between to elements of a semigroup. –  B. S. Feb 8 '13 at 6:03

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