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Kernel (algebra)
In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image. When the algebraic structures involved have an underlying group structure, the kernel is taken to be the preimage of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity. For example, the map that sends every integer to its parity (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers modulo 2, and its respective kernel would be the even integers which all have 0 as its parity. The kernel of a homomorphism of group-like structures will only contain the identity if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Two-sided Ideal
In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal. An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
