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ALGEBRA OVER A FIELD

Algebra over a field. An algebra (or linear algebra) over a field F is a ring R that is also a vector space with members of F as scalars and satisfies (ax)(by) = (ab)(xy) for all scalars a and b and all members x and y of R. The dimension of the vector space is the order of R. The algebra is a commutative algebra, or an algebra with unit element , according as the ring is a commutative ring, or a ring with unit element. A division algebra is an algebra that is also a division ring. A simple algebra is an algebra that is a simple ring.

James & James. Mathematics Dictionary

Examples.

1. The set of real numbers is a commutative division ring over the field of rational numbers.

2. The set of all square matrices of order n with complex numbers (or real numbers) as elements is a non-commutative algebra with identity over the field of real numbers.

3. The space of all linear operators on a vector space, with composition as the product, is an algebra with identity.

References

James & James. Mathematics Dictionary

Hoffman, Kunze. Linear Algebra. p. 108

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