Math Genius: Finite rings of prime order must have a multiplicative identity

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The standard definition of a ring is an abelian group that is a monoid under multiplication (with distributivity). However, there are some books that have a weaker definition implying that a ring only has to be closed under multiplication (no identity).

There is a problem in my algebra book asking me to prove that if a ring (defined in the second way) has $p$ elements, where $p$ is prime, and the multiplication is not trivial (i.e. sending everything to $0$), then the ring is forced to have a multiplicative identity.

Its seems like a trivial proof, but I just can’t see what I’m missing.

What I have so far:
Given $R$ is a ring with $p$ elements and
$R$ is an abelian group of prime order, therefore it is cyclically generated, and of characteristic $p$ and isomorphic to $mathbb{Z}/pmathbb{Z}$. Essentially it boils down to showing $mathbb{Z}/pmathbb{Z}$ is forced to have a multiplicative identity, but I just can’t see where this comes from (every resource I found seems to take this as a fact). Since this is a requirement regardless of multiplicative structure, I can’t just use the fact that $mathbb{Z}/pmathbb{Z} – { 0 }$ is a group under the typical multiplication.

Let $x$ be a nonzero element of the ring. Then $R={0,x,2x,3x,ldots,(p-1)x}$
where $2x$ means $x+x$ etc. Then $x^2=jx$ where $1le jle p-1$.
Moreover $(ax)(bx)=abx^2=(abk)x$. All you need to do is to prove that for
some $a$, $(abk)x=bx$ for all $b$. (It’s surely enough to do this for $b=1$).

If $R$ is a ring of order $p$, then it injects into the ring $text{End}(mathbb{Z}/pmathbb{Z})$ of endomorphisms of the abelian group $mathbb{Z}/pmathbb{Z}$. How many elements does this ring have?

Consider the subset $Z ⊆ R$ of elements such that left-multiplication by $z ∈ Z$ does map everything on $0$, $Z$ is trivial because $R$ has non trivial multiplication, hence every non zero element $a$ of $R$ defines an automorphism of the abelian group $R$ which sends $x$ to $ax$ (because the kernel of each map is trivial) , hence we can define a map from $Rsetminus{0}$ to the automorphisms group $operatorname{Aut}R$ this map is injective, or $operatorname{Aut}R$ has $p-1$ element then our map send a non zero element $a$ of $R$ on the identity morphism of $R$, we obtain $ax=x$ for every element $x$ of $R$.

Go on a hunt for all the ways in which R can fail to be isomorphic to ℤ/pℤ, and demonstrate the failure of any of them to hold.

Consider the subset Z ⊆ R of elements such that left-multiplication by z ∈ Z does map everything to zero. We may easily show that Z is an ideal in R:

  • For y,z ∈ Z, we have (y+z)r = yr + zr = 0+0 = 0 for any r ∈ R, so y+r ∈ Z;
  • For any z ∈ Z and a,r ∈ R, we have (az)r = a(zr) = 0 and (za)r = z(ar) = 0; so az, za ∈ Z.

Well, really, we only needed the first of these two criteria, because it shows that (Z,+) must be a subgroup of the abelian group (R,+), which is of prime order. We know that Z ≠ R; so it follows that Z = {0}.

Okay then. If none of the non-zero elements map everything to zero via multiplication, perhaps some of them map some things to zero? Select a ∈ R  {0} arbitrary, and consider the set Sa = {b ∈ R | ab = 0}. Well, again we find that Sa is an abelian subgroup of R, and again we know that Sa is not allowed to be all of R; so we know that Sa = {0}. That is, R contains no zero divisors.

Well, are there any non-zero elements such that multiplication is not invertible? Well, no: any a,b ∈ R such that ar = br for some non-zero r ∈ R would have the property (a − b)r = 0, so that a = b; so multiplication by elements of R  {0} yield bijections on R  {0}.

This should now enable you to answer your question.

$rm R = {0,a,2a,cdots,(p-1)a} $ for $rm 0 ne ain R:$. So $rm a^2 = k:a $ and $rm k ne 0:$ via multiplication nontrivial. Now $rm j = 1/k :mod p Rightarrow ia ja = ijka = ia Rightarrow ja = 1:$.

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