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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 group, and scalar ...
Localization (commutative algebra) In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions such that the denominator s belongs to a given subset S of R.
Nakayama's lemma. In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem [1] — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules. Informally, the lemma immediately gives a precise ...
Support of a module. In commutative algebra, the support of a module M over a commutative ring R is the set of all prime ideals of R such that (that is, the localization of M at is not equal to zero). [1] It is denoted by . The support is, by definition, a subset of the spectrum of R .
Conformal geometry. In mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a space. In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal ...
v. t. e. In mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A. This is thus an algebraic structure with an addition, a multiplication, and a scalar multiplication (the multiplication by the image of the ring homomorphism of an element of K ).
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 ...
A finite geometry is any geometric system that has only a finite number of points . The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry.
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