Abstract Nonsense

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 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.

June 19, 2011 -

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