simple algebra

Noun

Meaning

simple algebra (plural simple algebras)

(algebra) An algebra that contains no nontrivial proper (two-sided) ideals and whose multiplication operation is not zero (i.e., there exist a and b such that ab ≠ 0).

Examples

A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. Source: Internet

A central simple algebra over K is a matrix algebra over a (finite dimensional) division algebra with center K. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions. Source: Internet

For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. Source: Internet

In this section, a central simple algebra is assumed to have finite dimension. Source: Internet

Since the center of a simple k-algebra is a field, any simple k-algebra is a central simple algebra over its center. Source: Internet

The Skolem–Noether theorem states any automorphism of a central simple algebra is inner. Source: Internet

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