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The ring of integers of a number field K, denoted O_K, is the set of algebraic integers in K, which is a ring of dimension d over Z, where d is the extension degree of K over ...

Let (X,A,mu) and (Y,B,nu) be measure spaces. A measurable rectangle is a set of the form A×B for A in A and B in B.

In a lattice, any two elements a and b have a greatest lower bound. This greatest lower bound is often called the meet of a and b, and is denoted by a ^ b. One can also speak ...

A point that lies on one of the sides of a triangle but is a not vertex.

A group G such that the quotient group G/Z(G), where Z(G) is the group center of G, is Abelian. An equivalent condition is that the commutator subgroup G^' is contained in ...

A topology induced by the metric g defined on a metric space X. The open sets are all subsets that can be realized as the unions of open balls B(x_0,r)={x in X|g(x_0,x)<r}, ...

Let R be a number ring of degree n with 2s imaginary embeddings. Then every ideal class of R contains an ideal J such that ||J||<=(n!)/(n^n)(4/pi)^ssqrt(|disc(R)|), where ...

Two objects form a mirror pair if one can be translated and rotated in such a way that the two objects together possess mirror symmetry, i.e., one is the mirror image of the ...

A module homomorphism is a map f:M->N between modules over a ring R which preserves both the addition and the multiplication by scalars. In symbols this means that ...

The kernel of a module homomorphism f:M-->N is the set of all elements of M which are mapped to zero. It is the kernel of f as a homomorphism of additive groups, and is a ...