Division Ring and Integral Domain

Question

In abstract algebra, there are division ring, integral domain and field.

Division Ring is a Ring when all elements are unit.

Integral Domain is a Ring with no div '0' and commutative.

Is there any example of

1. Division Ring, but not I.D.

2. I.D but not Division Ring

I can't literally imagine any of them.

Only thing I can guess is if I.D. is finite, then it is a field so 2. is an infinite one.

Answer 1

1. A division ring is not necessarily an integral domain. An integral domain is a commutative ring by definition. A division ring is not necessarily commutative.

2. The set of integers is an integral domain which is not a division ring. For example, $2 x = 1$ has no solution for $x$ in the integers.

Thanks to Jyrki Lahtonen for correcting my initial answer.

But please note: apparently not all authorities demand that an integral domain is commutative. If you are using such a definition of integral domain, then yes, it is true that all division rings are also integral domains.

I will do some research to find out what sources use such a definition for an integral domain.

EDIT: Found it. According to Wikipedia:

• J.C. McConnell and J.C. Robson "Noncommutative Noetherian Rings" (Graduate Studies in Mathematics Vol. 30, AMS)

does not insist that an integral domain necessarily has to be commutative.

Answer 2

Units are no zero divisors. Let $u$ be a unit with $uv=0$. Then $v = 1 v = (u^{-1}u)v = u^{-1}(uv) = u^{-1}0=0$, since $0$ is absorbing.

Thus each (commutative) D.R. is an I.D.

Conversely, not every I.D. is a D.R.