Question about units and zero divisors in an arbitrary ring I know that in a finite commutative ring $R$ with identity, with $a\in R$, $a$ is either a unit or a zero divisor.  Also, in the case of an arbitrary ring, if $a\in R$, is a unit, then it cannot be a zero divisor.  For the converse, if $a$ is not a zero divisor then $a$ is not necessarily a unit.   Example would be the set of integers. 
So in general, whether an element in a ring is a units or a zero divisors can be summarized as follows:  
Any element in an arbitrary ring (finite, infinite, commutative, non-commutative, associative, non associate, with identity, without identity) if it is a zero divisor, then it is not a unit and conversely 
In a finite commutative ring with identity, if an element is a unit, then it is not a zero divisor and conversely.
What happens if we have the case of finite commutative ring without identity, would the statement still hold?
If an element in a ring with identity is a unit, then it is not a zero divisor
In the case of arbitrary ring with or without an identity, especially in the case of rings with infinite elements, if an element is a unit, then it can not be a zero divisor.  But for the converse an element is not a zero divisor might not necessarily be a unit. What if then we impose extra condition(s) (commutativity, associativity, identity, etc), would it be possible for elements in such a ring to be like in the case of finite commutative ring, to be either a unit or a zero divisor.
Thank you in advance
 A: 
Any element in an arbitrary ring [...] if it is a zero divisor, then it is not a unit and conversely

If it is a zero divisor, then it is not a unit, but not conversely.  It can be not a unit and not a zero divisor. Every element in $\mathbb Z\setminus\{0,1,-1\}$ is an example.

In a finite commutative ring with identity, if an element is a unit, then it is not a zero divisor and conversely.

True, but you don't even need to include commutativity for that.

What happens if we have the case of finite commutative ring without identity, would the statement still hold? "If an element in a ring with identity is a unit, then it is not a zero divisor"

As mentioned in the comments, there is no point in considering units in any ring that has no identity, because they are not defined.
There is a rather obscure condition on rings that goes like this:
A ring is called right cohopfian if every element that is not a left zero divisor is a unit. (The analogous left hand condition is defined similarly.)
If we consider any right-and-left cohopfian ring (with identity, otherwise it doesn't make sense) or, if you prefer, a commutative cohopfian ring, then the elements of the ring are partitioned into units and zero divisors.  That seems to be the most precise description of what you are talking about.
Maybe the most famous class of rings that are two-sided cohopfian are two-sided Artinian rings.
