Invariant basis number
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over R have a well-defined rank. In the case of fields, the IBN property becomes the statement that finite-dimensional vector spaces have a unique dimension.
A ring R has invariant basis number (IBN) if for all positive integers m and n, Rmisomorphic to Rn (as left R-modules) implies that m = n.
Equivalently, this means there do not exist distinct positive integers m and n such that Rm is isomorphic to Rn.
Rephrasing the definition of invariant basis number in terms of matrices, it says that, whenever A is an m-by-n matrix over R and B is an n-by-m matrix over R such that AB = I and BA = I, then m = n. This form reveals that the definition is left-right symmetric, so it makes no difference whether we define IBN in terms of left or right modules; the two definitions are equivalent.
Note that the isomorphisms in the definitions are not ring isomorphisms, they are module isomorphisms.
The main purpose of the invariant basis number condition is that free modules over an IBN ring satisfy an analogue of the dimension theorem for vector spaces: any two bases for a free module over an IBN ring have the same cardinality. Assuming the ultrafilter lemma (a strictly weaker form of the axiom of choice), this result is actually equivalent to the definition given here, and can be taken as an alternative definition.
...
Wikipedia