Well, that's the question: what type of elements are included in the non-units of a finite commutative ring? I know that there are zero divisors, but I don't know about other type of elements. In a finite ring, all non units elements are zero divisors?
In the case of the infinite commutative rings is very different? Is clear that for example in $\mathbb{Z}[x]$ there are elements that are non units and aren't zero divisors, like the element $x$, and of course, in $\mathbb{Z}_4\times \mathbb{Z}[x]$ there exists zero divisors, just as $(2,0)$. Are other type of non units in infinite rings?
EDIT: When I made this question was considering the fact that I'm talking about a ring that has non units. I understand the fact that integral domains doesn't have non units (Please forgive me for not specify this before P Vanchinathan)
Best Answer
There are rings without any zero-divisors. They are called integral domains. In finite rings, being integral domain is the same as being a field, so every non-zero element is a unit there.
But I strongly suggest you change the mental picture of a ring: your words what non-units are in a ring makes me think that your mental picture of a ring is one is full of units, and a few mutants called non-units. You should think of units as a few elements of rings that happen to have multiplicative inverses; ( we call rings where every nonzero element is a unit by a special name fields).
One classifies elements as zero divisors, units, and non-units. Then perhaps one talks of irreducible elements, prime elements (these are all non-units), but this means you are in a UFD. In Z, the non-units can be two kinds composites and primes. In polynomial rings, one can classify non-units as irreducibles and reducibles; also as elements of degree 1, elements of degree 359 etc (these are non-units).
The reason I emphasize this is that in Ring theory one mostly studies ideals of the rings: they are the counterpart of normal subgroups of group theory. And ideals do not contain any unit (well in fact it can contain a unit, and in that case it will be unique, namely the whole ring no a very interesting one).
EDIT: Please study rings along with homomorphisms and ideals. If one has a homomorphism defined on a ring almost always anything in the kernel is a non-unit.Take polynomials in many variables with real numbers as coefficients. Only the constants are units. And anything else is a non-unit.