Ideal (ring Theory) - Further Properties

Further Properties

• In rings with identity, an ideal is proper if and only if it does not contain 1 or equivalently it does not contin a unit.
• The set of ideals of any ring are partially ordered via subset inclusion, in fact they are additionally a complete modular lattice in this order with join operation given by addition of ideals and meet operation given by set intersection. The trivial ideals supply the least and greatest elements: the largest ideal is the entire ring, and the smallest ideal is the zero ideal. The lattice is not, in general, a distributive lattice.
• Unfortunately Zorn's lemma does not necessarily apply to the collection of proper ideals of R. However when R has identity 1, this collection can be reexpressed as "the collection of ideals which do not contain 1". It can be checked that Zorn's lemma now applies to this collection, and consequently there are maximal proper ideals of R. With a little more work, it can be shown that every proper ideal is contained in a maximal ideal. See Krull's theorem at maximal ideal.
• The ring R can be considered as a left module over itself, and the left ideals of R are then seen as the submodules of this module. Similarly, the right ideals are submodules of R as a right module over itself, and the two-sided ideals are submodules of R as a bimodule over itself. If R is commutative, then all three sorts of module are the same, just as all three sorts of ideal are the same.
• Every ideal is a pseudo-ring.
• The ideals of a ring form a semiring (with identity element R) under addition and multiplication of ideals.