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An operation on rings and modules. Given a commutative unit ring R, and a subset S of R, closed under multiplication, such that 1 in S, and 0 not in S, the localization of R ...

A module is a mathematical object in which things can be added together commutatively by multiplying coefficients and in which most of the rules of manipulating vectors hold. ...

Let K be a field of arbitrary characteristic. Let v:K->R union {infty} be defined by the following properties: 1. v(x)=infty<=>x=0, 2. v(xy)=v(x)+v(y) forall x,y in K, and 3. ...

Noncommutative topology is a recent program having important and deep applications in several branches of mathematics and mathematical physics. Because every commutative ...

By asking a small number of innocent-sounding questions about an unknown number, it is possible to reconstruct the number with absolute certainty (assuming that the questions ...

The concept of a space is an extremely general and important mathematical construct. Members of the space obey certain addition properties. Spaces which have been ...

A technical mathematical object defined in terms of a polynomial ring of n variables over a field k. Syzygies occur in tensors at rank 5, 7, 8, and all higher ranks, and play ...

A group action G×X->X is transitive if it possesses only a single group orbit, i.e., for every pair of elements x and y, there is a group element g such that gx=y. In this ...

A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space R^n, where every element ...

The derivative of a function represents an infinitesimal change in the function with respect to one of its variables. The "simple" derivative of a function f with respect to ...