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 algebra_over_a_commutative_ring [2013/08/31 17:27]nikolaj algebra_over_a_commutative_ring [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/08/31 17:45 ben 2013/08/31 17:27 nikolaj 2013/08/31 17:22 nikolaj 2013/08/31 17:21 nikolaj created Next revision Previous revision 2013/08/31 17:45 ben 2013/08/31 17:27 nikolaj 2013/08/31 17:22 nikolaj 2013/08/31 17:21 nikolaj created Line 1: Line 1: ===== 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: $[\cdot,​\cdot] : A\times A\to A$ | | $x,y,z\in A$ | | $x,y,z\in A$ | Line 12: Line 12: Bilinearity:​ Bilinearity:​ - | @#55EE55: $[r\cdot x+s\cdot y,z]=r\cdot [x,​z]+s\cdot [y,z]$ | + | @#55EE55: postulate ​  | @#55EE55: $[r\cdot x+s\cdot y,z]=r\cdot [x,​z]+s\cdot [y,z]$ | - | @#55EE55: $[z,r\cdot x+s\cdot y]=r\cdot [z,​x]+s\cdot [z,y]$ | + | @#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]]