I'm looking for some insight to a question on modules defined over non-commuative rings.
Let $M$ be a module of finite rank over some ring $R$, with $N\leq M$ a submodule.
If $R$ is commutative, it is stated in the post Rank of free module and its free submodule that we always have $Rk(N)\leq Rk(M)$ when $R$ is an integral domain (with a deceptively tricky proof).
I have looked a little around and haven't found any answers for the same question where $R$ is non-commutative. This post from Overflow directed me to The Theory of Rings, but the terminology used there applies to non-commutative principal ideal domains (and no mention is made with regards to this specific question about submodules).
I understand that difficulty may arise from the fact that in general we may not have a well-defined field of fractions in the non-commutative case, as indicated in the afore-referenced post.
My question: if $R$ is a non-commutative domain (meaning if $ab=0$ in $R$ then either $a=0$ or $b=0$), does the inequality $Rk(N)\leq Rk(M)$ still hold?
Thanks.
P.S: apologies for the edits. I was having some trouble getting the roman TeX to format properly.
