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)

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    $\begingroup$ Your edit in the question shows you haven't got the point in my answer. No need to say a ring with non-units, as if it is a pathological case. It sounds like saying I live in a house that has a wall. $\endgroup$ – P Vanchinathan Nov 3 '16 at 15:13
  • $\begingroup$ I understood your answer, you change the point of view of the different ones are the units and I understand it, but I ask more than that, for example the question In a finite ring, all non units elements are zero divisors? because I wanted to know a contradiction of the question or if it's right. $\endgroup$ – MonsieurGalois Nov 3 '16 at 15:17
  • $\begingroup$ For your question about finite rings, see math.stackexchange.com/questions/60969/…. $\endgroup$ – Eric Wofsey Nov 9 '16 at 22:36
  • $\begingroup$ It's really unclear what you mean by "type". Every element of a ring is either a zero divisor or not a zero divisor; what more do you want? $\endgroup$ – Eric Wofsey Nov 9 '16 at 22:38
  • $\begingroup$ In the case of a finite ring I wanted to know if there are other elements than the zero divisors, but I have proved that there are only zero divisors. In the infinite case I wanted to know how much different is from the case of finite rings. Is clear that there could be any type of element, but I wanted to know if having for example a zero divisor or a idempotent element it could avoid having other type of elements. $\endgroup$ – MonsieurGalois Nov 9 '16 at 22:52

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.

  • $\begingroup$ I understand the idea that a ring could not have zero divisors or even non unit elements (fields for example), but I mean: In the case that a ring has non units in it, what type of elements could have? $\endgroup$ – MonsieurGalois Nov 3 '16 at 14:57
  • $\begingroup$ @MonsieurGalois See the paragraph I added in my answer. $\endgroup$ – P Vanchinathan Nov 3 '16 at 15:03
  • $\begingroup$ I understand your answer, but I can't see why this answers the question. I know that $\mathbb{R}[x]$ is a ring in which all constant elements are units and the others are non, but I asked for the type of elements included in the non-units, so this is only a case where all elements are units. As in the last example I added there are cases where there are zero divisors, so I want to know what other type of non units in a ring could exist? (in the finite and infinite case) $\endgroup$ – MonsieurGalois Nov 3 '16 at 15:08

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