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Ideal operations

The sum and product of ideals are defined as follows. For a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}}, left (resp. right) ideals of a ring R, their sum is

which is a left (resp. right) ideal,and, if a, b {\displaystyle {\mathfrak {a}},{\mathfrak {b}}} are two-sided,

i.e. the product is the ideal generated by all products of the form ab with a in a {\displaystyle {\mathfrak {a}}} and b in b {\displaystyle {\mathfrak {b}}}.

Note a + b {\displaystyle {\mathfrak {a}}+{\mathfrak {b}}} is the smallest left (resp. right) ideal containing both a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}} (or the union a ∪ b {\displaystyle {\mathfrak {a}}\cup {\mathfrak {b}}}), while the product a b {\displaystyle {\mathfrak {a}}{\mathfrak {b}}} is contained in the intersection of a {\displaystyle {\mathfrak {a}}} and b {\displaystyle {\mathfrak {b}}}.

The distributive law holds for two-sided ideals a, b, c {\displaystyle {\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}},

If a product is replaced by an intersection, a partial distributive law holds:

where the equality holds if a {\displaystyle {\mathfrak {a}}} contains b {\displaystyle {\mathfrak {b}}} or c {\displaystyle {\mathfrak {c}}}.

Remark: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a complete modular lattice. The lattice is not, in general, a distributive lattice. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a quantale.

If a, b {\displaystyle {\mathfrak {a}},{\mathfrak {b}}} are ideals of a commutative ring R, then a ∩ b = a b {\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}} in the following two cases (at least)

(More generally, the difference between a product and an intersection of ideals is measured by the Tor functor: Tor 1 R (R / a, R / b) = (a ∩ b) / a b {\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}}.)

An integral domain is called a Dedekind domain if for each pair of ideals a ⊂ b {\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}}}, there is an ideal c {\displaystyle {\mathfrak {c}}} such that a = b c {\displaystyle {\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}}. It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the fundamental theorem of arithmetic.