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. $\mathfrak{a\subseteq b\quad\Rightarrow\quad ac\subseteq bc}$ endenumerate

   Remark. The properties 1, 2, 3, 4 together with the properties $$\mathfrak{(a+b)+c = a+(b+c),\quad a+b = b+a,\quad a}+(0) = \mathfrak{a}$$ 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).