## Characteristic of a Ring (Pt. II)

**Point of Post: **This is a continuation of this post.

With this characterization we have the following fascinating theorem:

**Theorem: ***Let and be unital rings with unities and . Then, every unital homomorphism induces a unital epimomorphism .*

**Proof: **It’s evident that we may restrict to a unital morphism , thus it suffices to prove that . To do this we merely note that is a unital homomorphism and thus we may conclude from prior theorem that . Thus,

and so the conclusion follows.

From this we get the following result which is useful in proving there doesn’t exist unital homomorphisms from a ring to a ring :

**Theorem: ***Let and be unital rings. If there exists a unital homomorphism then .*

**Proof: **Assume first that , then we have by definition that is infinite, and since by the previous theorem we may conclude that is infinite, and thus , and since this case holds. Suppose then that . If we are done since , so assume not, so that . We know then from basic group theory that since is a group epimorphism and are finite that or . The conclusion follows.

This allows one to conclude many interesting things about the existence of unital morphisms between two unital rings, things like there is no unital homomorphism between .

*Characteristics for Non-Unital Rings*

We now seek to define the notion of characteristic for non-unital rings which generalizes the notion of the characteristic for unital rings. To motivate this consider the following:

**Theorem: ***Let be a unital ring with unity . Then, if and only if for all and no smaller such number has this property. Moreover, if and only if there exists no such that for all .*

**Proof: **Suppose first that that . Then, for any one has that and if one has that and so clearly does not hold for every . Conversely, suppose that for every . Evidently then we have that . Note then that for each there exists some such that and thus otherwise . Thus, .

Suppose then that , then for every we have that and so does not hold for every . Conversely, suppose that then that for every there exists some such that . Clearly then since otherwise .

Thus, it makes sense to generalize the notion of characteristic to non-unital rings by defining the *characteristic *of a (possibly non-unital) ring , denoted , to be if for all and no smaller natural number has that property and if there does not exists a natural number such that for all . This last theorem tells us that this notion of characteristic and the one previously defined agree on unital rings and thus it is unambiguous to write .

*Restrictions of Characteristic on Integral Domains and Division Rings*

What we now show is that the characteristic of some classes of rings are greatly restricted. In particular we shall show that for integral domains and division rings the characteristic is either or a prime. This makes a lot of sense since,well, it’s so intuitive that the intuition is the proof:

**Theorem: ***Let be a integral domain or a division ring, then is either zero or a prime.*

**Proof: **Suppose that , with . We then have that . That said, since we have by definition that , but this is impossible if is either a division ring or an integral domain. The conclusion follows.

as a corollary we obviously have that

**Theorem: ***Let be a field, then is either zero or prime.*

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

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