A nonzero ring has invariant basis number if as free modules implies for all positive integers . The property is not automatic for arbitrary rings; the commutative-ring case is proved below.
If is a nonzero commutative ring and , reduction modulo a maximal ideal produces isomorphic vector spaces of dimensions and . Hence .
Articles by others on the same topic
The Invariant Basis Number (IBN) is a concept associated with the study of vector spaces and modules in abstract algebra, particularly in the context of infinite-dimensional vector spaces or modules over a ring. The invariant basis number of a vector space or a module refers to the property that, regardless of the choice of basis, the cardinality of the basis remains the same.