ideal multiplication laws

The multiplication (http://planetmath.org/ProductOfIdeals) 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 a 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

\mathfrak{(a+b)+c=a+(b+c),\qquad a+b=b+a,\qquad 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 (http://planetmath.org/Ring).

References

1 M. Larsen & P. McCarthy: Multiplicative theory of ideals. Academic Press, New York (1971).

Title	ideal multiplication laws
Canonical name	IdealMultiplicationLaws
Date of creation	2014-05-11 17:05:52
Last modified on	2014-05-11 17:05:52
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	16
Author	pahio (2872)
Entry type	Definition
Classification	msc 16D25
Synonym	laws of ideal product
Related topic	DivisibilityInRings
Related topic	ProductOfLeftAndRightIdeal
Related topic	InvertibilityOfRegularlyGeneratedIdeal