**Example 1**. In the category of sets let be a set and a collection of subsets of We also have iff The map is defined to be the inclusion map. Clearly is a direct system.

**Example 2**. Let be a commutative ring and let be a non-empty subset of such that no element of is nilpotent. The partial order is defined on by iff in i.e. for some For every let be the localization of at We need now to define the morphisms If then for some and so we can define by for all It is easy to see that is a ring homomorphism. Clearly is the identity map over because in this case we may choose Also, if with and then and hence

Thus and so is a direct system in the category of rings.

**Example 3**. Let be a ring and let be a two-sided ideal of Let be the set of positive integers and put for every For every the ring homomorphism is defined naturally by This ring homomorphisms are well-defined because if then Clearly is the identity map of and if then

Thus is an inverse system in the category of rings.

Similarly, if is an -module and we put then we will have the inverse system in the category of -modules. Here is defined by whenever

**Example 4**. Let be a group and let be a family of normal subgroups of which have finite index in We define the partial order on by if and only if We also define, whenever the group homomorphism by It is easily seen that is an inverse system in the category of finite groups.

**Example 5**. Let be a commutative ring, let be an -module and let be a direct system of -modules over some partially ordered index set For every let and for all with define by That means for all and

Claim . is a direct system of -modules.

*Proof*. So we need to prove two things to show that is a direct system:

1) for all This is obvious because and hence, for every and we have

2) whenever This also follows very easily from

How do you show that the partial order in Example 2 is antisymmetric?

You are absolutely right, it is not antisymmetric but we do not need this property of posets to define a direct system (and direct limit). We only need our index set to be reflexive and transitive. Although this is clear, but still I should have mentioned it when I was defining direct systems. We do need elements of to be non-nilpotent in order to be able to localize