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).