Difference between revisions of "Algebra (structure)"
(New page: Let <math>R</math> be a commutative ring. We say that a set <math>E</math> is an '''<math>R</math>-algebra''' if <math>E</math> is an <math>R</math>-module and we have a <math>A</...) |
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Let <math>R</math> be a [[commutative ring]]. We say that a set <math>E</math> is an '''<math>R</math>-algebra''' | Let <math>R</math> be a [[commutative ring]]. We say that a set <math>E</math> is an '''<math>R</math>-algebra''' | ||
| − | if <math>E</math> is an <math>R</math>-[[module]] and we have a <math> | + | if <math>E</math> is an <math>R</math>-[[module]] and we have a <math>R</math>-[[bilinear mapping]] of <math>E\times E</math> into |
<math>E</math>, denoted multiplicatively. That is, we have a multiplication between elements of <math>E</math>, | <math>E</math>, denoted multiplicatively. That is, we have a multiplication between elements of <math>E</math>, | ||
| − | and between elements of <math> | + | and between elements of <math>R</math> and elements of <math>E</math> such that for any <math>r \in R</math>, <math>x,y \in E</math>, |
<cmath> r(xy) = (rx)y = x(ry) , </cmath> | <cmath> r(xy) = (rx)y = x(ry) , </cmath> | ||
and | and | ||
<cmath> r(x+y) = rx + ry. </cmath> | <cmath> r(x+y) = rx + ry. </cmath> | ||
| − | We identify elements <math>r</math> of <math> | + | We identify elements <math>r</math> of <math>R</math> with the corresponding elements <math>r1</math> of <math>E</math>. |
Note that multiplication in <math>E</math> need not be [[associative]] or [[commutative]]; however, | Note that multiplication in <math>E</math> need not be [[associative]] or [[commutative]]; however, | ||
| − | the elements of <math> | + | the elements of <math>R</math> must commute and associate with all elements of <math>E</math>. We can thus think |
| − | of <math>E</math> as an <math> | + | of <math>E</math> as an <math>R</math>-module endowed with a certain kind of multiplication. |
Equivalently, we can say that <math>E</math> | Equivalently, we can say that <math>E</math> | ||
| − | is an <math> | + | is an <math>R</math>-algebra if it is a not-necessarily-associative ring that contains <math>R</math> as a sub-ring. |
{{stub}} | {{stub}} | ||
Latest revision as of 17:19, 8 September 2009
Let 
be a commutative ring. We say that a set 
is an -algebra
if is an -module and we have a -bilinear mapping of 
into
, denoted multiplicatively. That is, we have a multiplication between elements of ,
and between elements of and elements of such that for any 
, 
,
![\[r(xy) = (rx)y = x(ry) ,\]](http://latex.artofproblemsolving.com/e/a/2/ea27d317813f88117250ccc42d15e2714ea6a444.png)
and
![\[r(x+y) = rx + ry.\]](http://latex.artofproblemsolving.com/e/f/0/ef05a4497c7f89e8fa4cb841a4c207b37d506d0c.png)
We identify elements 
of with the corresponding elements 
of .
Note that multiplication in need not be associative or commutative; however,
the elements of must commute and associate with all elements of . We can thus think
of as an -module endowed with a certain kind of multiplication.
Equivalently, we can say that
is an -algebra if it is a not-necessarily-associative ring that contains as a sub-ring.
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