Prime elements in a noncommutative ring

Is there a reasonable definition of prime element in a noncommutative ring? The definition from wikipedia makes the assumption of commutativity and I'd like to know how necessary this condition is. Given a ring $R$, define:

We say $p$ is an right prime in $R$ iff for all $x,y,k\in R$, $kp=xy\to\exists m\in R,mp\in\{x,y\}$. We say $p$ is a left prime in $R$ if it is a right prime in $R^{op}$, and $p$ is a two-sided prime if it is both left and right prime.

Note that the condition of being right prime is the same as $p\mid xy\to p\mid x\lor p\mid y$, where $|$ is the right divisibility relation (I spelled it out for clarity). Obviously all three conditions are equivalent to being a prime in $R$ if $R$ is commutative. Also, $p$ is right prime iff the right principal ideal $(p)_r$ generated by $p$ is a prime ideal. (Not certain if $p$ is two-sided prime iff the two-sided principal ideal is a prime ideal.)

Is there a reason such a definition is not popular? Does it inherit any of the properties of primes in commutative rings? I'm not really sure what nontrivial theorems you can say about primes though in an arbitrary commutative ring - most of the interesting math seems to come from stronger conditions that use primes in their definitions, i.e. "every element is a product of primes" (factorization domain) or "every prime ideal contains a prime" (UFD), which always assume that $R$ is commutative anyway.

The first place to start exploring this would be Cohn's Noncommutative unique factorization domains.

He defines a primes more like "irreducible elements," and defines noncommutative associates, and obtains some nice results.

Beware though: if I remember right there is an error in the paper. But I also think I remember that the error was not a central result, and did not damage the central results. (It was more of a comment in passing that was wrong.)