Abstract
A vector space is called a linear algebra when besides addition of vectors and multiplication of vectors by scalars, a third operation, namely, multiplication of vectors by vectors is also defined in the vector space. In a finite dimensional linear associative algebra, multiplication is distributive with respect to addition. An algebra whose multiplication table is the operational table of a group is called “group algebra.” An algebra whose multiplication table is the operational table of a finite semi-group is called “semi-group algebra.” An algebra has no non-zero idempotent element if every element of the algebra is nilpotent. This chapter focuses on subsets of algebras. The one dimensional algebra of real numbers and the two dimensional algebra of complex numbers are the commutative division algebras over the field of real numbers. However, the four dimensional algebra of quaternion numbers is the non-commutative division algebra over the field of real numbers.
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