PlanetMath: every ring is an integer algebra

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every ring is an integer algebra

(Example)

Let be a ring. Then is also an algebra over the ring of integers if we define the action of $\mathbb{Z}$ on by the following rules:

$\displaystyle 0 \cdot x = 0$

$\displaystyle (n + 1) \cdot x = n \cdot x + x$

$\displaystyle (-n) \cdot x = -(n \cdot x)$

In other words, the action of a positive integer

is to add

to itself

times and the action of a negative integer

is to subtract

to itself

times.

"every ring is an integer algebra" is owned by rspuzio.

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See Also: general associativity

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Cross-references: negative, integer, positive, words, action, ring of integers, algebra, ring
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This is version 2 of every ring is an integer algebra, born on 2004-11-04, modified 2006-12-25.
Object id is 6449, canonical name is EveryRingIsAnIntegerAlgebra.
Accessed 1450 times total.

Classification:

AMS MSC:	16S99 (Associative rings and algebras :: Rings and algebras arising under various constructions :: Miscellaneous)
	20C99 (Group theory and generalizations :: Representation theory of groups :: Miscellaneous)
	13B99 (Commutative rings and algebras :: Ring extensions and related topics :: Miscellaneous)

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