Abstract Nonsense

Crushing one theorem at a time

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.

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Motivation

Often when working with things like vector spaces over arbitrary fields etc. one is concerned with when 2=0. Namely, something will happen when a+a=0 and you’d like to conclude that a=0 (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 1+1=0? More generally, as the same sort of question often comes up, what is the smallest n for which n1=\underbrace{1+\cdots+1}_{n\text{ times}}=0? We shall see that there are multiple interesting ways of interpreting this problem and that the result is perhaps more surprising than you think.

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Characteristic of a Unital Ring

Let R be a unital ring with unity 1_R. We define the characteristic of R, denoted by \text{char}(R), as follows:

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\text{char}(R)=\begin{cases}m & \mbox{if}\quad |1_R|=m<\infty\\ 0 & \mbox{if}\quad |1_R|=\infty\end{cases}

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where |\cdot| denotes the order in the abelian group (R,+). More explicitly, we define \text{char}(R) to be m\in\mathbb{N} if

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m1_R=\underbrace{1_R+\cdots+1_R}_{m\text{ times}}=0

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and no integer smaller has this property and we define it to be 0 if m1_R\ne 0 for every m\in\mathbb{N}. For example, the rings \mathbb{Z}_2, \mathbb{Z}_2[x] (polynomials with coefficients in \mathbb{Z}_2), and \mathbb{Z}_2^n all have characteristic 2 while the rings \mathbb{Z}, \mathbb{Q}, \mathbb{R}[x], and C(\mathbb{R}) all have characteristic 0. It’s easy to see that having characteristic zero implies that your ring is infinite while the converse is not true.

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What we’d first like to do is come up with some alternate characterizations of the characteristic of a ring. Firstly:

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Theorem: Let R be a unital ring with unity 1_R. Then, there exists a unique unital homomorphism u_R:\mathbb{Z}\to R. Moreover, this homomorphism has the property that \ker u_R=\text{char}(R)\mathbb{Z} (all multiples of \text{char}(R)).

Proof: Let g,f be any two unital homomorphisms \mathbb{Z}\to R. Since f(1)=1_R=g(1) we see that f,g agree on a generating set of the group (\mathbb{Z},+) and so from basic group theory (since f is, if anything, a group homomorphism) we may conclude that f=g. Thus, it suffices to produce one unital morphisms \mathbb{Z}\to R. To do this define u_R(z)=zf(1_R). To verify that this is a unital morphism we note that u_R(1)=11_R=1_R, u_R(ab)=(ab)1_R=(a1_R)(b1_R)=u_R(a)u_R(b), and uR(a+b)=(a+b)1_R=a1_R+b1_R=u_R(a)+u_R(b) (this can all be easily verified by induction if this isn’t convincing enough) and so u_R is a unital morphism. We note then that since \ker u_R is a subring and  thus in particular a subgroup we know from the basics of cyclic group  theory that \ker u_R=m\mathbb{Z} for some m\in\mathbb{N}. Note then that since m\in m\mathbb{Z}=\ker u_R we have that 0=u_R(m)=m1_R and moreover since m is the minimum positive element of m\mathbb{Z}=\ker u_R we have that if 0<k<m that 0\ne u_R(k)=k1_R. Thus, it follows that m=\text{char}(R) and the theorem follows. \blacksquare

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Remark: For those who know the lingo, \mathbb{Z} is an initial object in the category \bold{Ring} of unital rings.

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In fact, it’s easy to see from the above that we can then define the characteristic of R to be equal to the cardinality of \text{im }u_R (for this above unital morphism \mathbb{Z}\to R) if \#(\text{im }u_R)<\infty and 0 otherwise.

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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 2\mathbb{Z} is a subring of the ring \mathbb{Z} it is not a unital subring since it doesn’t contain 1). 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 1 (so that their intersection does as well). We then define the smallest unital subring or prime subring of a ring R to be the intersection of all unital subrings of R. 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 R_\cap. What we claim is that the characteristic of R is the cardinality of R_\cap if it’s finite and 0 otherwise. Moreover, (as this previous statement should probably have tipped you off to) it’s true that R_\cap=\text{im }u_R. This makes logical sense since intuitively (and truthfully) the smallest subring is just integer polynomial combinations of 0 and 1 and since for 1 addition and multiplication are the ‘same thing’ the subring generated by those two will just be equal to the subgroup generated by \{1\}.

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Theorem: Let R be a unital ring with unity 1_R and u_R the unique unital homomorphism \mathbb{Z}\to R. Then, R_\cap=\text{im }u_R=\langle 1_R\rangle (where \langle\cdot\rangle denotes subgroup generated by).

Proof: To prove that R_\cap=\text{im }u_R we note that since \text{im }u_R is a unital subring (it’s a subring and it contains 1_R) we have that R_\cap\subseteq\text{im }u_R. Conversely, if a\in\text{im }u_R then a=z1_R for some z\in\mathbb{Z} and since R_\cap is a subring, and in particular a subgroup, containing 1_R we must have that a=z1_R\in R_\cap. Thus, \text{im }u_R=R_\cap as desired. Now, to prove that \text{im }u_R=\langle 1_R\rangle we note that since \text{im }u_R is a subring, and in particular a subgroup, containing 1_R we must have that \langle 1_R\rangle\subseteq\text{im }u_R. Conversely, if a\in\text{im }u_R then a=z1_R for some z\in\mathbb{Z} and so by definition z\in\langle1_R\rangle and so \text{im }u_R=\langle 1_R\rangle as desired. \blacksquare

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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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June 19, 2011 - Posted by | Algebra, Ring Theory | , , , , , , , ,

4 Comments »

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