10

Let $(B, +, \cdot)$ be a non-trivial ring with the property that every $x \in B$ satisfies $x \cdot x = x$. How does one prove that such a ring $(B, +, \cdot)$ must have a unit element $1_B$? (Or, in case this is not true in general, what is a counterexample?)

BTW, I'm looking for an elementary proof, not requiring anything more than the definition of a ring, the definition of $(B, +, \cdot)$, and, if necessary, the easily shown facts that $x + x = 0$ and $x\cdot y = y\cdot x,\,\forall\, x,y \in B$.

kjo
  • 14,904
  • 1
    My definition of ring includes a $1$, and my definition of boolean algebra also includes a $1$. – Zhen Lin May 14 '13 at 08:35
  • 1
    @ZhenLin I learned it as "ring" and then "ring with unity". But whatever the definition, I think it's clear that this post in particular is referring to rings without a 1. – Caleb Stanford May 14 '13 at 08:44
  • @ZhenLin: What do you call something that is like $2\mathbb{Z}$, which is similar to what you call a "ring", except that it does not include a unit? I will gladly rephrase my question to use your terminology. – kjo May 14 '13 at 08:49
  • I wonder, can anyone produce an example of a Boolean rng as a proper ideal of a Boolean ring? This is the "obvious" thing to try until one runs into the problem that these ideals often have identities (which is not the identity of the original ring). The larger Boolean ring must necessarily be infinite. – Ragib Zaman May 14 '13 at 09:12
  • @RagibZaman Unless I did something silly I think that's easy :) Please review my solution. – rschwieb May 15 '13 at 13:28

4 Answers4

14

An example: the family of all finite subsets of a given infinite set.

Boris Novikov
  • 17,754
  • 2
    @kjo: Let your ground set be $\mathbb{N}$ and the elements be binary sequences, where a $1$ in the $n$ coordinate means that $n$ is in your set, and a zero means that is not. Then define addition and multiplication coordinate wise, and work modulo $2$. Then the unit would have to be $(1,1,1,1...)$ which will fail to be in your ring since you are consider finite families. – Daniel Montealegre May 14 '13 at 08:53
  • 3
    I really think you should be more explicit than this (although the questioner just accepted this answer...) – Rhys May 14 '13 at 08:57
  • 1
    Yes, the ring operations should be specified. Or just take $\oplus_n \mathbb{F}_2$, as in Daniel's comment. – Martin Brandenburg May 14 '13 at 09:13
  • 3
    I think the operations are evident and natural. – Boris Novikov May 14 '13 at 09:19
  • 7
    @BorisNovikov: I don't think so. I happen to know some operations that work (namely, symmetric difference for $+$ and intersection for $\cdot$), and that's why I decided to accept your answer (reluctantly), but I consider them neither "evident" nor "natural". – kjo May 14 '13 at 09:30
  • @kjo: OK, sorry. Next time I will write more rigorously :-) – Boris Novikov May 14 '13 at 10:24
  • I think this is isomorphic somehow to the solution I gave (my solution is the newcomer, to be clear), but when you express it as vectors with binary entries, it's a bit easier to grasp the operations. – rschwieb May 15 '13 at 13:30
  • @rschwieb Anyway our answer seems interesting to me. – Boris Novikov May 16 '13 at 08:27
  • @BorisNovikov For sure :) Thanks for posting it! – rschwieb May 16 '13 at 10:50
7

In the ring $\prod_{i=1}^\infty\Bbb Z_2$, consider the ideal $\oplus_{i=1}^\infty\Bbb Z_2$. It is a subrng without identity.

(Ultimately I think this is isomorphic to Boris' example, but here the operations are clear.)

rschwieb
  • 160,592
4

Here is a positive result: Let $R$ be a commutative ring such that every element can be written as a sum of products of elements of $R$ (i.e. the multiplication map $R \otimes_{\mathbb{Z}}R \to R$ is surjective). If $R$ is finite, then $R$ is unital. In particular, every finite boolean ring is unital (which also can be proven directly, of course).

Proof: Consider the unitalization $R^+$ as an $R$-module. Then we have $R R = R$, hence by Nakayama there is some $e \in R$ with $(1-e) R=0$. But this means that $e$ is a unit of $R$.

nilo de roock
  • 2,855
Martin Brandenburg
  • 181,922
  • Thanks for your answer. It flies so many miles above my head that I can't reasonably ask you to explain it further (my question requested elementary proofs for a reason). Needless to say, I can't even begin to evaluate it, but others may find it useful. – kjo May 14 '13 at 09:38
  • Yes, and one day, perhaps you will find it useful too. ;) – Martin Brandenburg May 14 '13 at 09:48
  • @YACP He did use it: it would not be possible to write the $1$ in $(1-e)R$ otherwise. – rschwieb May 16 '13 at 10:38
  • @MartinBrandenburg Looks like you meant rather "consider $R$ as an $R^+$ module," rather than the other way around, and then $M=R\lhd R^+$ puts you on the road to a nice application of Nakayama's lemma :) – rschwieb May 16 '13 at 11:31
  • Using "by Nakayama" is not helping, but showing off. – nilo de roock Oct 10 '20 at 08:32
0

Claim: A finite Boolean ring $B$ must have an identity.

Define a binary operation called "union" on $B$. $a\cup b = a+b+ab\tag*{}$ This operation is associative and commutative. And a good property of this operation is that $a(a\cup b)=a$ (gives the vibes of identity?). To prove this, use distributivity, idempotence and the fact that $x+x=0$ for all $x\in B$.

I would want to show that if $B=\{b_1,\ldots, b_n\}$ is your Boolean ring then its identity is $x=b_1\cup b_2\cup \ldots\cup b_n$.

Why is this the identity?

Hint: Here's my working with $b_1$. You can do this with each $b_i$ as well since $\cup $ is commutative and associative.

$b_1x\\ = b_1(b_1\cup b_2\cup\ldots\cup b_n) \\ = b_1(b_1\cup c) \quad\text{where $c=b_2\cup\ldots\cup b_n$}\\ = b_1$

In case $B$ is infinite, can you argue in this way that $B$ must have identity ? Answer is no. Think on this!

Nothing special
  • 3,690
  • 1
  • 7
  • 27