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Given a nonzero finitely generated module M over a commutative Noetherian local ring R with maximal ideal M and a proper ideal I of R, the Hilbert-Samuel function of M with ...

Given a short exact sequence of modules 0->A->B->C->0, (1) let ...->P_2->^(d_2)P_1->^(d_1)P_0->^(d_0)A->0 (2) ...->Q_2->^(f_2)Q_1->^(f_1)Q_0->^(f_0)C->0 (3) be projective ...

The length of all composition series of a module M. According to the Jordan-Hölder theorem for modules, if M has any composition series, then all such series are equivalent. ...

The natural projection, also called the homomorphism, is a logical way of mapping an algebraic structure onto its quotient structures. The natural projection pi is defined ...

Let F,G:C->D be functors between categories C and D. A natural transformation Phi from F to G consists of a family Phi_C:F(C)->G(C) of morphisms in D which are indexed by the ...

There are several meanings of "null vector" in mathematics. 1. The most common meaning of null vector is the n-dimensional vector 0 of length 0. i.e., the vector with n ...

The parallelogram law gives the rule for vector addition of vectors A and B. The sum A+B of the vectors is obtained by placing them head to tail and drawing the vector from ...

A vector perpendicular to a given vector a is a vector a^_|_ (voiced "a-perp") such that a and a^_|_ form a right angle. In the plane, there are two vectors perpendicular to ...

There are two different definitions of "polar vector." In elementary math, the term "polar vector" is used to refer to a representation of a vector as a vector magnitude ...

A map defined by one or more polynomials. Given a field K, a polynomial map is a map f:K^n->K^m such that for all points (x_1,...,x_n) in K^n, ...