# ideal multiplication laws

The multiplication (http://planetmath.org/ProductOfIdeals) of the (two-sided) ideals of any ring $R$ has following properties:

1. 1.

$(0)\mathfrak{a=a}(0)=(0)$

2. 2.

$\mathfrak{(ab)c=a(bc)}$

3. 3.

$\mathfrak{a(b+c)=ab+ac,\quad(a+b)c=ac+bc}$

4. 4.

If $R$ has a unity, then  $R\mathfrak{a}=\mathfrak{a}R=\mathfrak{a}$

5. 5.

If $R$ is commutative, then  $\mathfrak{ab=ba}$

6. 6.

$\mathfrak{ab\subseteq a\cap b}$

7. 7.

$\mathfrak{a(b\cap c)\subseteq ab\cap ac}$

8. 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 IdealMultiplicationLaws 2014-05-11 17:05:52 2014-05-11 17:05:52 pahio (2872) pahio (2872) 16 pahio (2872) Definition msc 16D25 laws of ideal product DivisibilityInRings ProductOfLeftAndRightIdeal InvertibilityOfRegularlyGeneratedIdeal