On Nov 6, 9:39 pm, junoexpress <mtbrenne...@gmail.com> wrote:
> Hi,
> I am trying to get an idea about how to classify rings based on the > various properties they can have. > In this regard, I am stumped by the question of whether there exist > rings which are:
> Case 1) Non-commutative and have no-zero divisors: I know that Z and > Z_p (p prime) have no zero divisors, but they are commutative. The > only non-commutative rings I can really think of are matrices, but > they can have zero divisors. (In fact, I've tried to either consider > subrings of matrices which might not have zero divisors or construct > various rings based on vectors, like a ring whose elements are vectors > in R^2 or R^3, where + is the typical componentwise addition and * is > either a dot or cross-product. While the cross product, is non- > commutative, it still has zero divisors). My guess is that these types > of structures exist though, but I just don't know enough examples of > rings to identify one that satisfies these conditions.
What happened to the quaternions? The quaternions with integer coefficients, if you want to avoid skew-fields...
> Case 2) Have unit elements, but are nor groups under *, not > commutative, and have no-zero divisors: No clues as to if this could > even exist. These conditions seem very restrictive and I have to > wonder if this type of structure even exists.
The quaternions with integer coefficients, for one.
And yes, they do exist.
If you have a ring with no zero divisors in which every nonzero element is invertible, you have what is called a "skew-field" or a "division ring". One reason you might be having trouble coming up with examples is that there are no finite division rings that are not fields (that is, not commutative); this is called Wedderburn's Theorem.
A ring with no zero divisors and a unit, but possibly non-commutative, is sometimes called an "entire ring" (careful, though; some authors, e.g. Lang, use that term for integral domains).
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