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algebra_over_a_commutative_ring [2013/08/31 17:22] nikolaj |
algebra_over_a_commutative_ring [2014/03/21 11:11] (current) |
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| ===== Algebra over a commutative ring ===== | ===== Algebra over a commutative ring ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $A$...R-module | | + | | @#55CCEE: context | @#55CCEE: $A$...R-module | |
| - | | @#55EE55: $\langle A,[\cdot,\cdot]\rangle \in \mathrm{algebra}(A)$ | | + | | @#55EE55: postulate | @#55EE55: $\langle A,[\cdot,\cdot]\rangle \in \mathrm{algebra}(A)$ | |
| - | | @#88DDEE: $[\cdot,\cdot] : A\times A\to A$ | | + | | @#55CCEE: context | @#55CCEE: $[\cdot,\cdot] : A\times A\to A$ | |
| | $x,y,z\in A$ | | | $x,y,z\in A$ | | ||
| | $r,s\in R$ | | | $r,s\in R$ | | ||
| - | | @#55EE55: $[r\cdot x+s\cdot y,z]=r\cdot [x,z]+s\cdot [y,z]$ | | + | Bilinearity: |
| - | | @#55EE55: $[z,r\cdot x+s\cdot y]=r\cdot [z,x]+s\cdot [z,y]$ | | + | |
| + | | @#55EE55: postulate | @#55EE55: $[r\cdot x+s\cdot y,z]=r\cdot [x,z]+s\cdot [y,z]$ | | ||
| + | | @#55EE55: postulate | @#55EE55: $[z,r\cdot x+s\cdot y]=r\cdot [z,x]+s\cdot [z,y]$ | | ||
| ==== Discussion ==== | ==== Discussion ==== | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[https://en.wikipedia.org/wiki/Algebra_%28ring_theory%29|Algebra (Ring theory)]] | Wikipedia: [[https://en.wikipedia.org/wiki/Algebra_%28ring_theory%29|Algebra (Ring theory)]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| === Refinement of === | === Refinement of === | ||
| - | [[module]] | + | [[Module]] |
