algebraic categories without free objects
An initial object is always a free object. So in the context of algebraic systems with a trivial object, such as groups, or modules, there is always at least one free object. However, we usually dismiss this example as it does not lead to any useful results such as the existence of presentations.
However, there are many ways in which a cateogry of algebraic objects can fail to include non-trivial free objects.
1 Restriction to finite sets
Similarly, finite modules of in a module category over an infinite ring are never free. For examples use the rings , , etc.
However, this is not always the case. For example, if we consider finite -modules (vector spaces) each of these are free.
2 Homomorphism restrictions
In the category of rings with 1 it is often beneficial to force all ring homomorphisms to be unital. However, this restriction can prevent the construction of free objects.
Suppose is a free ring in the category of rings with positive characteristic. Then we ask, what is the characteristic of ?
If it is then we choose another ring of a different characteristic, a characteristic relatively prime to , and then there can be no unital homomorphism from to . So must have characteristic 0. In contrast to the above examples we have not excluded infinite objects in this restriction. This example is even more powerful than those above as it also exclude the existance of an initial object, so indeed NO free objects exist in this category.
|Title||algebraic categories without free objects|
|Date of creation||2013-03-22 16:51:25|
|Last modified on||2013-03-22 16:51:25|
|Last modified by||Algeboy (12884)|