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An ideal I of a ring R is called principal if there is an element a of R such that I=aR={ar:r in R}. In other words, the ideal is generated by the element a. For example, the ...

A Cartesian product equipped with a "product topology" is called a product space (or product topological space, or direct product).

The projective general linear group PGL_n(q) is the group obtained from the general linear group GL_n(q) on factoring by the scalar matrices contained in that group.

The projective general orthogonal group PGO_n(q) is the group obtained from the general orthogonal group GO_n(q) on factoring the scalar matrices contained in that group.

The projective general unitary group PGU_n(q) is the group obtained from the general unitary group GU_n(q) on factoring the scalar matrices contained in that group.

The projective symplectic group PSp_n(q) is the group obtained from the symplectic group Sp_n(q) on factoring by the scalar matrices contained in that group. PSp_(2m)(q) is ...

Given a vector space V, its projectivization P(V), sometimes written P(V-0), is the set of equivalence classes x∼lambdax for any lambda!=0 in V-0. For example, complex ...

A proper subgroup is a proper subset H of group elements of a group G that satisfies the four group requirements. "H is a proper subgroup of G" is written H subset G. The ...

A subvariety of an algebraic variety that is not the entire variety. The proper subvarieties of a line are its nonempty finite subsets, the proper subvarieties of a plane are ...

A metric space Z^^ in which the closure of a congruence class B(j,m) is the corresponding congruence class {x in Z^^|x=j (mod m)}.