For the domain of a linear map, this property equals the sum of its value for the kernel and for the image, by the rank-nullity theorem.
One of these properties for a commutative ring is given by the supremum of the length of chains of prime ideals. Another one of these is defined as the minimal value x such that every open cover admits a refining open cover with no point contained in more than x +1 sets. By using the Steinitz exchange lemma, the well-definedness of another one of these properties can be proven by (*) swapping vectors between sets. For the domain of a linear map, this property equals the sum of its value for the kernel and for the image, by the rank-nullity theorem. Rn is isomorphic to any real vector space with a value of n for this property, given by the size of a basis. For 10 points, give this property which for Euclidean space how many coordinates are needed to specify a point.
One of these properties for a commutative ring is given by the supremum of the length of chains of prime ideals.
dimension