Algebra over a field

:This article is about a particular kind of vector space. For other uses of the term "algebra" see algebra (disambiguation).

Index-free notation

The definition of a k-algebra is also frequently given in index-free notation. In this case, a k-algebra is a field K and a ring A together with a ring morphism

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:eta:K o A

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Since η is a ring morphism, then one must have either that A is the trivial ring, or that η is injective. This definition is equivalent to that above, with scalar multiplication

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:K imes A o A

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

given by

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:ka mapsto eta(k) a.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~