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2. Length of a module - Wikipedia

   en.wikipedia.org/wiki/Length_of_a_module

   In algebra, the length of a module over a ring is a generalization of the dimension of a vector space which measures its size. [ 1] page 153 It is defined to be the length of the longest chain of submodules. For vector spaces (modules over a field), the length equals the dimension. If is an algebra over a field , the length of a module is at ...

3. Measure (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Measure_(mathematics)

   Measure (mathematics) Informally, a measure has the property of being monotone in the sense that if is a subset of the measure of is less than or equal to the measure of Furthermore, the measure of the empty set is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.

4. Interval (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Interval_(mathematics)

   The diameter may be called the length, width, measure, range, or size of the interval. The size of unbounded intervals is usually defined as +∞, and the size of the empty interval may be defined as 0 (or left undefined). The centre of a bounded interval with endpoints a and b is (a + b)/2, and its radius is the half-length | a − b |/2 ...

5. Resolution (algebra) - Wikipedia

   en.wikipedia.org/wiki/Resolution_(algebra)

   Resolution (algebra) In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution[ 1]) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object ...

6. Module (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Module_(mathematics)

   Module (mathematics) In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module also generalizes the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers . Like a vector space, a module is an additive abelian ...

7. Metric space - Wikipedia

   en.wikipedia.org/wiki/Metric_space

   An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points (1, 0) and (-1, 0) can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between ...