definition of algebra over field
Topics
About: algebra over field
The table of contents of this article
Starting Context
- The reader knows a definition of %field name% vectors space.
Target Context
- The reader will have a definition of algebra over field.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\(*A\): \(\in \{\text{ the vectors spaces over } F\}\), with any multiplication, \(\bullet: A \times A \to A\), specified below
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Conditions:
\(\forall r_1, r_2, r'_1, r'_2 \in F, \forall v_1, v_2, v'_1, v'_2 \in A ((r_1 v_1 + r_2 v_2) \bullet (r'_1 v'_1 + r'_2 v'_2) = (r_1 r'_1) (v_1 \bullet v'_1) + (r_1 r'_2) (v_1 \bullet v'_2) + (r_2 r'_1) (v_2 \bullet v'_1) + (r_2 r'_2) (v_2 \bullet v'_2))\)
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2: Note
In other words, \(\bullet\) is bilinear.
Inevitably, for each \(v \in A\), \(v \bullet 0 = 0 \bullet v = 0\), because \(v \bullet 0 = (v + 0) \bullet (0 + 0) = (1 v + 0 v) \bullet (0 v + 0 v) = (1 0) (v \bullet v) + (1 0) (v \bullet v) + (0 0) (v \bullet v) + (0 0) (v \bullet v) = 0 (v \bullet v) + 0 (v \bullet v) + 0 (v \bullet v) + 0 (v \bullet v) = 0 + 0 + 0 + 0 = 0\) and \(0 \bullet v = (0 + 0) \bullet (v + 0) = (0 v + 0 v) \bullet (1 v + 0 v) = (0 1) (v \bullet v) + (0 0) (v \bullet v) + (0 1) (v \bullet v) + (0 0) (v \bullet v) = 0 (v \bullet v) + 0 (v \bullet v) + 0 (v \bullet v) + 0 (v \bullet v) = 0 + 0 + 0 + 0 = 0\).
So, the requirement of this definition equals this set of conditions: 1) \((v_1 + v_2) \bullet v'_1 = v_1 \bullet v'_1 + v_2 \bullet v'_1\); 2) \(v_1 \bullet (v'_1 + v'_2) = v_1 \bullet v'_1 + v_1 \bullet v'_2\); 3) \(r_1 v_1 \bullet r'_1 v'_1 = (r_1 r'_1) (v_1 \bullet v'_1)\): supposing the condition of this proposition, 1) \((v_1 + v_2) \bullet v'_1 = (1 v_1 + 1 v_2) \bullet (1 v'_1 + 0) = 1 v_1 \bullet v'_1 + 1 v_1 \bullet 0 + 1 v_2 \bullet v'_1 + 1 v_2 \bullet 0 = 1 v_1 \bullet v'_1 + 0 + 1 v_2 \bullet v'_1 + 0 = v_1 \bullet v'_1 + v_2 \bullet v'_1\); 2) \(v_1 \bullet (v'_1 + v'_2) = (1 v_1 + 0) \bullet (1 v'_1 + 1 v'_2) = 1 v_1 \bullet v'_1 + 1 v_1 \bullet v'_2 + 0 \bullet 1 v'_1 + 0 \bullet 1 v'_2 = 1 v_1 \bullet v'_1 + 1 v_1 \bullet v'_2 + 0 + 0 = v_1 \bullet v'_1 + v_1 \bullet v'_2\); 3) \(r_1 v_1 \bullet r'_1 v'_1 = (r_1 v_1 + 0) \bullet (r'_1 v'_1 + 0) = (r_1 r'_1) (v_1 \bullet v'_1) + r_1 v_1 \bullet 0 + 0 \bullet r'_1 v'_1 + 0 \bullet 0 = (r_1 r'_1) (v_1 \bullet v'_1) + 0 + 0 + 0 = (r_1 r'_1) (v_1 \bullet v'_1)\); supposing the conditions, 1), 2), and 3), \((r_1 v_1 + r_2 v_2) \bullet (r'_1 v'_1 + r'_2 v'_2) = r_1 v_1 \bullet (r'_1 v'_1 + r'_2 v'_2) + r_2 v_2 \bullet (r'_1 v'_1 + r'_2 v'_2) = r_1 v_1 \bullet r'_1 v'_1 + r_1 v_1 \bullet r'_2 v'_2 + r_2 v_2 \bullet r'_1 v'_1 + r_2 v_2 \bullet r'_2 v'_2 = (r_1 r'_1) (v_1 \bullet v'_1) + (r_1 r'_2) (v_1 \bullet v'_2) + (r_2 r'_1) (v_2 \bullet v'_1) + (r_2 r'_2) (v_2 \bullet v'_2)\).
This definition does not require the associativity of the multiplication, which another definition requires. The reason is that if the associativity was required, "Lie algebra" would be a false name.
An algebra over a field may not be any ring, because the associativity may not hold and any multiplicative identity, \(1\), may not exist in it.
Any algebra satisfying the associativity is called "associative algebra", which some people call "algebra".
Any algebra with \(1\) is called "unitary algebra".
Any unitary associative algebra is a ring.
