Characteristic of a Ring (Pt. I)
Point of Post: In this post we give several different characterizations of the characteristic of a unital ring and prove some varying theorems about the characteristic.
Often when working with things like vector spaces over arbitrary fields etc. one is concerned with when . Namely, something will happen when and you’d like to conclude that (assuming you’re working in an integral domain or something of the sort). So, the question really reduces to asking in a unital ring whether ? More generally, as the same sort of question often comes up, what is the smallest for which ? We shall see that there are multiple interesting ways of interpreting this problem and that the result is perhaps more surprising than you think.
Characteristic of a Unital Ring
Let be a unital ring with unity . We define the characteristic of , denoted by , as follows:
where denotes the order in the abelian group . More explicitly, we define to be if
and no integer smaller has this property and we define it to be if for every . For example, the rings , (polynomials with coefficients in ), and all have characteristic while the rings , , , and all have characteristic . It’s easy to see that having characteristic zero implies that your ring is infinite while the converse is not true.
What we’d first like to do is come up with some alternate characterizations of the characteristic of a ring. Firstly:
Theorem: Let be a unital ring with unity . Then, there exists a unique unital homomorphism . Moreover, this homomorphism has the property that (all multiples of ).
Proof: Let be any two unital homomorphisms . Since we see that agree on a generating set of the group and so from basic group theory (since is, if anything, a group homomorphism) we may conclude that . Thus, it suffices to produce one unital morphisms . To do this define . To verify that this is a unital morphism we note that , , and (this can all be easily verified by induction if this isn’t convincing enough) and so is a unital morphism. We note then that since is a subring and thus in particular a subgroup we know from the basics of cyclic group theory that for some . Note then that since we have that and moreover since is the minimum positive element of we have that if that . Thus, it follows that and the theorem follows.
Remark: For those who know the lingo, is an initial object in the category of unital rings.
In fact, it’s easy to see from the above that we can then define the characteristic of to be equal to the cardinality of (for this above unital morphism ) if and otherwise.
Despite the above definitions being very concrete the most common definition of characteristic is involved in the notion of smallest unital subring. In particular, a unital subring is a subring of a unital ring which contains the unity of the ambient ring (e.g. while is a subring of the ring it is not a unital subring since it doesn’t contain ). It is clear that the arbitrary intersection of a unital subrings are unital subrings since we know that the intersections of subrings are subrings and each of the intersecting sets contains (so that their intersection does as well). We then define the smallest unital subring or prime subring of a ring to be the intersection of all unital subrings of . It is evidently smallest in the sense that it is the minimal element in the poset of unital subrings with inclusion. We denote this smallest subring by . What we claim is that the characteristic of is the cardinality of if it’s finite and otherwise. Moreover, (as this previous statement should probably have tipped you off to) it’s true that . This makes logical sense since intuitively (and truthfully) the smallest subring is just integer polynomial combinations of and and since for addition and multiplication are the ‘same thing’ the subring generated by those two will just be equal to the subgroup generated by .
Theorem: Let be a unital ring with unity and the unique unital homomorphism . Then, (where denotes subgroup generated by).
Proof: To prove that we note that since is a unital subring (it’s a subring and it contains ) we have that . Conversely, if then for some and since is a subring, and in particular a subgroup, containing we must have that . Thus, as desired. Now, to prove that we note that since is a subring, and in particular a subgroup, containing we must have that . Conversely, if then for some and so by definition and so as desired.
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.