If X is a topological space, there is a natural way of transforming X /~ into a topological space ; see quotient space for the details.

If N is a closed normal subgroup of a profinite group G, then the factor group G / N is profinite ; the topology arising from the profiniteness agrees with the quotient topology.

If I is a right ideal of R, then R / I is simple if and only if I is a maximal right ideal: If M is a non-zero proper submodule of R / I, then the preimage of M under the quotient map is a right ideal which is not equal to R and which properly contains I.

If a finite difference is divided by b − a, one gets a difference quotient.

If H is a subgroup of G, the set of left or right cosets G / H is a topological space when given the quotient topology ( the finest topology on G / H which makes the natural projection q: G → G / H continuous ).

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

If R is a commutative ring, and M is an R-module, we define the Krull dimension of M to be the Krull dimension of the quotient of R making M a faithful module.

If N is the nilradical of commutative ring R, then the quotient ring R / N has no nilpotent elements.

If we add the relation x < sup > 2 </ sup > = 1 to the presentation of Dic < sub > n </ sub > one obtains a presentation of the dihedral group Dih < sub > 2n </ sub >, so the quotient group Dic < sub > n </ sub >/< x < sup > 2 </ sup >> is isomorphic to Dih < sub > n </ sub >.

If X is a diffeological space and ~ is some equivalence relation on X, then the quotient set X /~ has the diffeology generated by all compositions of plots of X with the projection from X to X /~.

If e < sub > 1 </ sub >, ... e < sub > d </ sub > is a basis of V, the unital zero algebra is the quotient of the polynomial ring k ..., E < sub > n </ sub > by the ideal generated by the E < sub > i </ sub > E < sub > j </ sub > for every pair ( i, j ).

If we try to use the quotient to compute f '( 0 ), however, an undefined value will result, since | x | is nondifferentiable at x = 0.

If N is a normal subgroup of G, then the index of N in G is also equal to the order of the quotient group G / N, since this is defined in terms of a group structure on the set of cosets of N in G.

If G and H are finite groups, then the index of H in G is equal to the quotient of the orders of the two groups:

If ΔP is infinitesimal, then the difference quotient is a derivative, otherwise it is a divided difference:

If is the quotient map then it is a covering since the action of Z on C generated by is properly discontinuous.

If p is a regular cover, then Aut ( p ) is naturally isomorphic to a quotient of.

If G is not simply connected, then the lattice P ( G ) is smaller than P ( g ) and their quotient is isomorphic to the fundamental group of G.

If it is not, there are three possible problems: the multiplication is wrong, the subtraction is wrong, or a greater quotient is needed.

If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

If R is a quotient of by a homogeneous ideal, then the canonical surjection induces the closed immersion

If X and Y are algebraic structures of some fixed type ( such as groups, rings, or vector spaces ), and if the function f from X to Y is a homomorphism, then ker f will be a subalgebra of the direct product X × X. Subalgebras of X × X that are also equivalence relations ( called congruence relations ) are important in abstract algebra, because they define the most general notion of quotient algebra.

If G is an extension of Q by N, then G is a group, N is a normal subgroup of G and the quotient group G / N is isomorphic to group Q.

If X is the plane with the origin missing, and G is the infinite cyclic group generated by ( x, y )→( 2x, y / 2 ) then this action is wandering but not properly discontinuous, and the quotient space is non-Hausdorff.

If a particular object did not support a, it could be easily added in the module.

If M is an R module and is its ring of endomorphisms, then if and only if there is a unique idempotent e in E such that and.

If a is an idempotent of the endomorphism ring End < sub > R </ sub >( M ), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f < sup > 2 </ sup > is the identity endomorphism of M.

If r represents an arbitrary element of R, f can be viewed as a right R-homomorphism so that, or f can also be viewed as a left R module homomorphism, where.

If M is a free module over a principal ideal domain R, then every submodule of M is again free.

If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.

If k is a field and G is a group, then a group representation of G is a left module over the group ring k. The simple k modules are also known as irreducible representations.

If M is a module which has a non-zero proper submodule N, then there is a short exact sequence

* If L is a maximal left ideal, then R / L is a simple left R module.

The latter example leads to a generalization of modules over rings: If C is a preadditive category, then Mod ( C ) := Add ( C, Ab ) is called the module category over C. When C is the one-object preadditive category corresponding to the ring R, this reduces to the ordinary category of ( left ) R-modules.

* If a module is simple, then its endomorphism ring is a division ring ( this is sometimes called Schur's lemma ).

If the module is an injective module, then indecomposability is equivalent to the endomorphism ring being a local ring.

If the module is Artinian, Noetherian, projective or injective, then the endomorphism ring has a unique maximal ideal, so that it is a local ring.

* If an R module is finitely generated and projective ( that is, a progenerator ), then the endomorphism ring of the module and R share all Morita invariant properties.

If we interpret the object as the left module, then this matrix category becomes a subcategory of the category of left modules over.

If K is only a commutative ring and not a field, then the same process works if A is a free module over K. If it isn't, then the multiplication is still completely determined by its action on a set that spans A ; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism.

If H is a left module over the ring R, one forms the ( algebraic ) character module H * consisting of all abelian group homomorphisms from H to Q / Z.

If F ( U ) is a module over the ring O < sub > X </ sub >( U ) for every open set U in X, and the restriction maps are compatible with the module structure, then we call F an O < sub > X </ sub >- module.