Prof. Valdes and Prof. Anderson studied factorization in commutative rings with zero divisors. I was wondering how similar factorization is in anticommutative rings with zero divisiors.
$\begingroup$
$\endgroup$
9
-
$\begingroup$ Do you have a link to the article of Valdes & Anderson? Also do you really mean anti-commutative ($ab = -ab$) or 'just' non-commutative? $\endgroup$– VincentCommented Jul 12, 2022 at 14:26
-
1$\begingroup$ @Vincent I mean anticommutative. An example of anticommutative ring is R^3 under the cross product. $\endgroup$– Insulin69Commented Jul 12, 2022 at 14:31
-
$\begingroup$ researchgate.net/publication/… $\endgroup$– Insulin69Commented Jul 12, 2022 at 14:32
-
$\begingroup$ Okay. So they are naturally riddled with zero divisors as $x^2 = 0$ for all $x$ in such a ring. The most prominent examples are probably Lie algebras and exterior algebras $\bigwedge V$ for some vector space $V$. I would expect people have looked at factorization in those examples at least, but I don't know a reference $\endgroup$– VincentCommented Jul 12, 2022 at 14:33
-
2$\begingroup$ @Vincent Lie algebras aren't associative, so it's not clear what factorization would mean in that context (but maybe that's a good notion) and $\bigwedge V$ is not anticommutative in this strong sense because even wedges commute. $\endgroup$– Will SawinCommented Jul 12, 2022 at 18:12
|
Show 4 more comments