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[parent] ideal multiplication laws (Definition)

The multiplication of the (two-sided) ideals of any ring $ R$ has following properties:

  1. $ (0)\mathfrak{a = a}(0) = (0)$
  2. $ \mathfrak{(ab)c = a(bc)}$
  3. $ \mathfrak{a(b+c) = ab+ac, \quad (a+b)c = ac+bc}$
  4. If $ R$ has unity, then $ R\mathfrak{a} = \mathfrak{a}R = \mathfrak{a}$
  5. If $ R$ is commutative, then $ \mathfrak{ab = ba}$
  6. $ \mathfrak{ab \subseteq a\cap b}$
  7. $ \mathfrak{a(b\cap c) \subseteq ab\cap ac}$
  8. $ \mathfrak{a\subseteq b\quad\Rightarrow\quad ac\subseteq bc}$

Remark. The properties 1, 2, 3, 4 together with the properties

$\displaystyle \mathfrak{(a+b)+c = a+(b+c),\quad a+b = b+a,\quad a}+(0) = \mathfrak{a}$
of the ideal addition make the set $ A$ of all ideals of $ R$ to a semiring $ (A,\,+,\,\cdot)$. It is not a ring, since no non-zero ideal of $ R$ has the additive inverse.

Bibliography

1
M. LARSEN & P. MCCARTHY: Multiplicative theory of ideals. Academic Press. New York (1971).



"ideal multiplication laws" is owned by pahio.
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See Also: divisibility in rings, product of left and right ideal, invertibility of regularly generated ideal

Other names:  laws of ideal product
Keywords:  ideal product, ideal multiplication

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Cross-references: semiring, addition, commutative, unity, properties, ring, ideals
There are 3 references to this entry.

This is version 10 of ideal multiplication laws, born on 2006-03-11, modified 2008-01-14.
Object id is 7711, canonical name is IdealMultiplicationLaws.
Accessed 1876 times total.

Classification:
AMS MSC: 16D25 (Associative rings and algebras :: Modules, bimodules and ideals :: Ideals)
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