# Finite rings and subrings isomorphic to $\mathbb{Z}_n$

My book has proven this:

Every ring with unity has a subring isomorphic to either $\mathbb{Z}$ or $\mathbb{Z_n}$.

The $\mathbb{Z_n}$ case arises if the parent ring has characteristic $r>0$

I have this question: is it possible for a ring with characteristic $r>0$ to have two subrings isomorphic to $\mathbb{Z_n}$ and $\mathbb{Z_m}$ with $n \neq m$?

I believe so, yes. Consider $\mathbb{Z_8}$. clearly $\mathbb{Z_6} \subset \mathbb{Z_8}$ and $\mathbb{Z_6} \cong \mathbb{Z_6}$, and $\mathbb{Z_4} \subset \mathbb{Z_8}$ and $\mathbb{Z_4} \cong \mathbb{Z_4}$.

My question is how to prove this.

• Note that although $\{0,1,\dots,5\} \subset \mathbb{Z}_8$, we cannot say that $\mathbb{Z}_6$ is a subring of $\mathbb{Z}_8$. – Ben Grossmann Mar 12 '14 at 5:27
• Does your definition of subring (check your textbook) require that the subset includes the multiplicative identity? The answer to your question depends largely on this. – Ben Grossmann Mar 12 '14 at 5:32
• Can you explain why $\mathbb{Z_6}$ is not a subring of $\mathbb{Z_8}$? – terrible at math Mar 12 '14 at 5:43
• A subring is a subset of the elements of a ring that uses the addition and multiplication table of the original ring. Let $[5]$ and $[1]$ be the (obvious) elements in $\mathbb{Z}_8$. According to the addition table in $\mathbb{Z}_8$, $[5]+[1] = [6]$. Since this isn't in $\{[0],[1],[2],[3],[4],[5]\}$, those elements are not a subring of $\mathbb{Z}/8$. – Eric Towers Mar 12 '14 at 5:49
• Oh damn, thank you. That makes perfect sense, I should have realized this... Okay, what if we take a subring without unity? – terrible at math Mar 12 '14 at 5:52

user49685 got there first and should get credit for his work. (I've upvoted his answer...)

That a subring must contain the additive identity and the multiplicative identity requires that all subrings contain the characteristic subring. (I.e., they must contain the subring generated by $0$ and $1$, and since $0$ isn't doing any heavy lifting there, is the subring generated by $1$ -- the characteristic subring.) If such a subring contains an element not in the characteristic subring, then it is not singly generated, so is not cyclic.

Yes. $\mathbb{Z}/6\mathbb{Z}$ contains an isomorphic copy of $\mathbb{Z}/2\mathbb{Z}$ and an isomorphic copy of $\mathbb{Z}/3\mathbb{Z}$. This idea works for any ring with non-prime order.

Since subrings contain $1$, the method above does not work. However, any of $\mathbb{Z}_p \times \mathbb{Z}_q$, $\mathbb{Z}_p \oplus \mathbb{Z}_q$, $\prod_{i=2}^n \mathbb{Z}_i$, or $\bigoplus_{i=2}^n\mathbb{Z}_i$, would work rather transparently. (Need to be careful in the product or direct summation that only a finite number of distinct finite characteristics appear in the list of multiplicands or summands, otherwise the resulting object has characteristic zero. If, however, the "finite spectrum of characteristics" condition is satisfied, the product or direct sum can be extended to an infinite index set.)

Update to the update: This is still wrong. Too many habits from the category of rings that may or may not contain $1$.

• This depends on whether OPs definition of a subring requires containing the ring's multiplicative identity. – Ben Grossmann Mar 12 '14 at 5:36
• @Omnomnomnom: This is true. Been thinking in the category of rings that may or may not contain $1$ lately ... – Eric Towers Mar 12 '14 at 5:39
• Yes, the subrings must have unity as well. – terrible at math Mar 12 '14 at 5:41
• why doesn't it work if the subrings contain 1? – terrible at math Mar 12 '14 at 5:47
• @terribleatmath: I wouldn't sweat it now. It's another way to glue rings together. In overview, for finite index sets, $\bigoplus$ and $\prod$ are the same; for infinite sets, $\bigoplus$ is the set of elements of $\prod$ that have only a finite set of elements different from zero. $(1,1,1,\dots)$ is in the product, but not the direct sum. It is intuitively a way to model something like convergent series (although, to an algebraist, convergence is not the important part of these objects). (Even better, it models points in a Hilbert space finitely far from the origin.) – Eric Towers Mar 12 '14 at 6:14

If your subrings are required to contained the unit element of the former ring, then the answer is NO.

Let $X$ be any ring, and let $A, B$ are its subrings, assume that $A \cong \mathbb{Z}_n$, and $B \cong \mathbb{Z}_m$, where $n \neq m$. From $A \cong \mathbb{Z}_n$, we can deduce that the unit element $e$ of $X$ has additive order $n$ (note that $e \in A$ will be mapped to $\overline{1} \in \mathbb{Z}_n$); whereas the fact that $B \cong \mathbb{Z}_m$ yields the unit element $e$ of $X$ has additive order $m$, which is a contradiction.

If $p$ and $q$ are coprime and greater than 1, $\mathbb Z_{pq}$ does contain a copy of $\mathbb Z_q$.

Proof: Bezout's identity says that $$ap-bq=1$$ for some positive integers $a,b$ and $a<q$. Say that $\mathbb Z_{pq}=\{0,1,\ldots,pq-1\}$. We shall see that the subset $A=\{0,ap,2ap,(q-1)ap\}$ is a subring. The products are computed in $\mathbb Z_{pq}$, of course.

$A$ is clearly a subgroup, and as a group is isomorphic to $\mathbb Z_q=\{0,1,\ldots,q-1\}$, the isomorphism being the function $f(k\cdot ap)=k$. So we must show that $f$ is also an isomorphism of rings. Since $$f(jap\cdot kap)=f(jkap\cdot ap)=f(jk(bq+1)\cdot ap)=f(jkabpq+jkap)=f(jk\cdot ap),$$ $f$ is an isomorphism of rings.

The statement doesn't hold if we remove the condition that $p$ and $q$ are coprime. Take $\mathbb Z_8$. If it contained a (proper) subring, there would be an element $x\neq 0,1$ such that $x^2=x$, that is, $x(x-1)=0$. Since $x$ or $x-1$ are odd, the other must be multiple of $8$, and that implies that $x=0$ or $x=1$, a contradiction.

• The subrng has a different unit: $\,ap\,$ vs. $\,1,\,$ so it is not a subring (assuming ring means ring with $1).$ – Bill Dubuque Mar 12 '14 at 13:09
• I have checked the definition of subring in the wikipedia, and Bill Dubuque's comment is right. But I still find this fact interesting. Should not these subsets have a name? – ajotatxe Mar 12 '14 at 16:12
• As I said, it is a subrng. – Bill Dubuque Mar 12 '14 at 16:16