0

Does a ring $R$ with left and right non-zero zero divisors always has a identity?

My definition of left and right non-zero divisors are as follows:

An element $a\in R$ is said to be a left-zero (or right-zero) divisor of $R$ if $\exists b\neq 0$ such that $ab=0$(or $ba=0$).

However, I don't really get how to prove that $R$ has an identity element.

I think we need to construct one for $R$ but even then, I don't understand how to start the process.

Any hints/help regarding this will be greatly appreciated.

Thomas Finley
  • 3,625

1 Answers1

3

This is false. Take $R=\{\bar{0},\bar{2}\}\subset \mathbb{Z}/4\mathbb{Z}$.

Then $R$ is a ring (but not a subring) for the operations of $\mathbb{Z}/4\mathbb{Z}$. It has also non trivial zero divisors: $\bar{2}.\bar{2}=\bar{0}$.

But $R$ has no identity element: $\bar{0}$ cannot be an identity since $\bar{0}.\bar{2}\neq\bar{2}$, and $\bar{2}$ cannot be an identity since $\bar{2}.\bar{2}\neq\bar{2}$,

GreginGre
  • 16,641
  • Perfect example. But why do you say it's "not a subring"? Does "subring" necessarily/definitionally contain the "$1$" of the ring? I'd appreciate it if you would clarify your convention here. :) – paul garrett Dec 16 '23 at 18:41
  • 1
    I find it contrasting to the way I am studying ring theory; I recall that a subring may or may not have an identity and even if it has one, it need not be the same as that of the parent ring... – Nothing special Dec 16 '23 at 18:46
  • 1
    This depends on the context. In a lecture on factorial rings, one assumes that all rings are commutative and have a unity, and every subring has a unity, which is the one from the ring. – Dietrich Burde Dec 16 '23 at 18:49
  • 1
    @paulgarrett: for me, by definition, a subring of a ring with identity contains the identity element . – GreginGre Dec 16 '23 at 18:53
  • Thanks to everyone for clarifying "subring". I kinda suspected as much... though by now I myself have apparently somewhat lost many "global" (as opposed to "local") conventions... :) – paul garrett Dec 16 '23 at 19:12