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The descending chain condition, commonly abbreviated "D.C.C.," is the dual notion of the ascending chain condition. The descending chain condition for a partially ordered set ...

A module over a unit ring R is called divisible if, for all r in R which are not zero divisors, every element m of M can be "divided" by r, in the sense that there is an ...

An exact sequence is a sequence of maps alpha_i:A_i->A_(i+1) (1) between a sequence of spaces A_i, which satisfies Im(alpha_i)=Ker(alpha_(i+1)), (2) where Im denotes the ...

A module M over a unit ring R is called flat iff the tensor product functor - tensor _RM (or, equivalently, the tensor product functor M tensor _R-) is an exact functor. For ...

Green's identities are a set of three vector derivative/integral identities which can be derived starting with the vector derivative identities del ·(psidel phi)=psidel ...

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 ...

Householder (1953) first considered the matrix that now bears his name in the first couple of pages of his book. A Householder matrix for a real vector v can be implemented ...

For an n×n matrix, let S denote any permutation e_1, e_2, ..., e_n of the set of numbers 1, 2, ..., n, and let chi^((lambda))(S) be the character of the symmetric group ...

A matrix used in the Jacobi transformation method of diagonalizing matrices. The Jacobi rotation matrix P_(pq) contains 1s along the diagonal, except for the two elements ...