Exterior Algebra

The k-th exterior algebra of an ( R , R ) 𝑅 𝑅 (R,R) Bimodule M 𝑀 M over a Ring M 𝑀 M is the quotient of the Tensor Algebra

Tk⁢(M)=⨁i=0k⨂j=0iM=R⊕M⊕(M⊗M)⊕(M⊗M⊗M)⁢⋯superscript𝑇𝑘𝑀superscriptsubscriptdirect-sum𝑖0𝑘superscriptsubscripttensor-product𝑗0𝑖𝑀direct-sum𝑅𝑀tensor-product𝑀𝑀tensor-product𝑀𝑀𝑀⋯\par T^{k}(M)=\bigoplus_{i=0}^{k}\bigotimes_{j=0}^{i}M=R\oplus M\oplus(M% \otimes M)\oplus(M\otimes M\otimes M)\cdots

by the Two-Sided Ideal generated by { x ⊗ x } tensor-product 𝑥 𝑥 \left\{\;x\otimes x\right\} .