correspondence is given by for all . This correspondence commutes with the processes of taking sums and intersections (i.e., is a lattice isomorphism between the lattice of submodules of and the lattice of submodules of that contain ). Universal algebra To generalise this to universal algebra, normal subgroups need to be replaced by congruence relations. A congruence on an algebra is an equivalence relation that forms a subalgebra of considered as an algebra with componentwise operations. One can make the set of equivalence classes into an algebra of the same type by defining the operations via representatives; this will be well-defined since is a subalgebra of . The resulting structure is the quotient algebra. Theorem A (universal algebra) Let be an algebra homomorphism. Then the image of is a subalgebra of , the relation given by (i.e. the kernel of ) is a congruence on , and the algebras and are isomorphic. (Note that in the case of a group, iff , so one recovers the notion of kernel used in group theory in this case.) Theorem B (universal algebra) Given an algebra , a subalgebra of , and a congruence on , let be the trace of in and the collection of equivalence classes that intersect . Then is a congruence on , is a subalgebra of , and the algebra is isomorphic to the algebra . Theorem C (universal algebra) Let be an algebra and two congruence relations on such that . Then is a congruence on , and is isomorphic to Theorem D (universal algebra) Let be an algebra and denote the set of all congruences on . The set is a complete lattice ordered by inclusion. If is a congruence and we denote by the set of all congruences that contain (i.e. is a principal filter in , moreover it is a sublattice), then the map is a lattice isomorphism. Note References Emmy Noether, Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern, Mathematische Annalen 96 (1927) pp. 26–61 Colin McLarty, "Emmy Noether's 'Set Theoretic' Topology: From Dedekind to the rise of functors". The Architecture of Modern Mathematics: Essays in history and philosophy (edited by Jeremy Gray and José Ferreirós), Oxford University Press (2006) pp. 211–35. Paul M. Cohn, Universal algebra, Chapter II.3 p. 57