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👉 In mathematics, a subalgebra of a given algebra is a subset that is closed under all operations defined on it. This means that any element in the subalgebra can be obtained by applying an operation defined on the algebra to another element. For example, if A is an algebra over a field F and B is a subalgebra of A, then B is also an algebra over F. The operations defined on B include addition (A + B) and multiplication (B x A).

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👉 In mathematics, a subalgebra is a subset of a larger algebra that contains all the elements that are functions of the elements in the original algebra. The term "subalgebraical" refers to the fact that the operations and properties of algebraic structures can be extended to include functions of those structures. This includes both addition and multiplication, as well as other algebraic operations like division or exponentiation.

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👉 In mathematics, a subalgebraic structure is an algebraic structure that contains a smallest algebraic extension. This means it can be defined as an algebraic structure with a smaller set of elements than the original structure itself. For example, in a ring, an element x belongs to the subring R[x] if it satisfies some polynomial equation or relation involving x and its coefficients. Similarly, in a field, an element y belongs to the subfield F(y) if it satisfies a polynomial

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