# algebraic categories without free objects

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

However, this is not always the case. For example, if we consider finite $\mathbb{Z}_{p}$-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 $F$ is a free ring in the category of rings with positive characteristic. Then we ask, what is the characteristic  of $F$?

If it is $m>0$ then we choose another ring $R$ of a different characteristic, a characteristic relatively prime to $m$, and then there can be no unital homomorphism         from $F$ to $R$. So $F$ 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.

If we return to the full category of unital rings we observe every ring is a $\mathbb{Z}$-algebra   we can use the free associative algebras $\mathbb{Z}\langle X\rangle$ does exist here.

Title algebraic categories without free objects AlgebraicCategoriesWithoutFreeObjects 2013-03-22 16:51:25 2013-03-22 16:51:25 Algeboy (12884) Algeboy (12884) 4 Algeboy (12884) Example msc 08B20