# 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 16:19, 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 , ,
and
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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