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Superalgebra
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===Even subalgebra=== Let ''A'' be a superalgebra over a commutative ring ''K''. The [[submodule]] ''A''<sub>0</sub>, consisting of all even elements, is closed under multiplication and contains the identity of ''A'' and therefore forms a [[subalgebra]] of ''A'', naturally called the '''even subalgebra'''. It forms an ordinary [[algebra (ring theory)|algebra]] over ''K''. The set of all odd elements ''A''<sub>1</sub> is an ''A''<sub>0</sub>-[[bimodule]] whose scalar multiplication is just multiplication in ''A''. The product in ''A'' equips ''A''<sub>1</sub> with a [[bilinear form]] :<math>\mu:A_1\otimes_{A_0}A_1 \to A_0</math> such that :<math>\mu(x\otimes y)\cdot z = x\cdot\mu(y\otimes z)</math> for all ''x'', ''y'', and ''z'' in ''A''<sub>1</sub>. This follows from the associativity of the product in ''A''.
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