Hi all, I'm trying to understand the structure of an "algebra over a field".
The definition I get from wolfram is
"Formally, an algebra is a vector space V over a field F with a multiplication."
So if I was to break down this down....
A vector space has two operations
+:V x V --> V (addition)
. :F x V --> V
and so if A was an algebra over F, would it mean it has three operations?
+:A x A --> A (addition)
. :A x V --> A
* : A x A --> A (multiplication)
Thank you for your reply
Further, I'm a bit confused about the second part of the definition that states,
"The multiplication must be distributive and, for every and must satisfy
The distributive part I understand, but is the following bit a 'general rule' and not for example, an associative or commutative law?
for all now if you in 1) and 2) put you'll get the distributivity law and if you, in 1), put you'll get also if you, in 2), put you'll get
so the "bilinearlity" condition gives us your condition, i.e. for all and conversely, if we have distributivity and your condition, then
we'll have bilinearity.
but al these don't help you that much to understand what an algebra over a field is. there's an equivalent definition of an algebra over a field, which is easier and gives a better view:
an algebra over a field is a ring (with unity) which contains a copy of in its center. this is very simple, isn't it? so, containing a copy of in the center means there exists an injective
ring homomorphism from to the center of now we actually can identify with its copy and so we may assume that then when multiplying elements of by an element
we can move anywhere we want because is in the center of this also explains