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Malcev algebras

In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special neutral element (the zero vector in the case of vector spaces, the identity element in the case of commutative groups, and the zero element in the case of rings or modules). The characteristic feature of a Malcev algebra is that we can recover the entire equivalence relation ker f from the equivalence class of the neutral element.

To be specific, let A and B be Malcev algebraic structures of a given type and let f be a homomorphism of that type from A to B. If e is the neutral element of B, then the kernel of f is the preimage of the singleton set { e }; that is, the subset of A consisting of all those elements of A that are mapped by f to the element e.The kernel is usually denoted ker f (or a variation). In symbols:

Since a Malcev algebra homomorphism preserves neutral elements, the identity element e of A must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set { e }.

The notion of ideal generalises to any Malcev algebra (as linear subspace in the case of vector spaces, normal subgroup in the case of groups, two-sided ideals in the case of rings, and submodule in the case of modules). It turns out that ker f is not a subalgebra of A, but it is an ideal.Then it makes sense to speak of the quotient algebra G / (ker f).The first isomorphism theorem for Malcev algebras states that this quotient algebra is naturally isomorphic to the image of f (which is a subalgebra of B).

The connection between this and the congruence relation for more general types of algebras is as follows.First, the kernel-as-an-ideal is the equivalence class of the neutral element e under the kernel-as-a-congruence. For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction for vector spaces, modules, and rings).Using this, elements a and b of A are equivalent under the kernel-as-a-congruence if and only if their quotient a / b is an element of the kernel-as-an-ideal.-by sanjay pk