In Algebraic Structures
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. Thus the coimage of f is a quotient algebra of X much as the image of f is a subalgebra of Y; and the bijection between them becomes an isomorphism in the algebraic sense as well (this is the most general form of the first isomorphism theorem in algebra). The use of kernels in this context is discussed further in the article Kernel (algebra).
Read more about this topic: Kernel (set Theory)
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