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The matrix operations of 1. Interchanging two rows or columns, 2. Adding a multiple of one row or column to another, 3. Multiplying any row or column by a nonzero element.

A representation phi of a group G is faithful if it is one-to-one, i.e., if phi(g)=phi(h) implies g=h for g,h in G. Equivalently, phi is faithful if phi(g)=I_n implies g=e, ...

The general orthogonal group GO_n(q,F) is the subgroup of all elements of the projective general linear group that fix the particular nonsingular quadratic form F. The ...

Let K be a number field with ring of integers R and let A be a nontrivial ideal of R. Then the ideal class of A, denoted [A], is the set of fractional ideals B such that ...

A proper ideal of a ring that is not the intersection of two ideals which properly contain it. In a principal ideal domain, the ideal I=<a> is irreducible iff a=0 or a is an ...

A ring in which the zero ideal is an irreducible ideal. Every integral domain R is irreducible since if I and J are two nonzero ideals of R, and a in I, b in J are nonzero ...

A negative matrix is a real or integer matrix (a)_(ij) for which each matrix element is a negative number, i.e., a_(ij)<0 for all i, j. Negative matrices are therefore a ...

A nonpositive matrix is a real or integer matrix (a)_(ij) for which each matrix element is a nonpositive number, i.e., a_(ij)<=0 for all i, j. Nonpositive matrices are ...

Every dense linear order complete set without endpoints having at most omega disjoint intervals is order isomorphic to the continuum of real numbers, where omega is the set ...

If a subset S of the elements of a field F satisfies the field axioms with the same operations of F, then S is called a subfield of F. In a finite field of field order p^n, ...