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[parent] strict divisibility (Definition)

Let $ a$ and $ b$ be elements of a commutative ring with non-zero unity and let $ a \mid b$. If there is a positive integer $ n$ such that $ a^n \mid b$ and $ a^{n+1} \nmid b$, then $ b$ is strictly divisible by $ a^n$; this may be denoted by

$\displaystyle a^n \parallel b.$
For example, $ 2^4 \parallel 48$.

Note. The expression “strictly divisible” may be used of course in a divisor monoid, too.



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"strict divisibility" is owned by pahio.
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See Also: divisibility, p-adic valuation, order valuation

Also defines:  strictly divisible
Keywords:  divisible

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Cross-references: monoid, divisor, expression, integer, positive, non-zero unity, commutative ring
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This is version 4 of strict divisibility, born on 2007-09-04, modified 2008-05-08.
Object id is 9916, canonical name is StrictDivisibility.
Accessed 314 times total.

Classification:
AMS MSC: 11A51 (Number theory :: Elementary number theory :: Factorization; primality)
 13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility)
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