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Alternating algebra
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<h2 id="examples">Examples</h2> <ul><li>The <a href="/facts/Differential_forms/T9gyHtMv">differential forms</a> on a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a> form an alternating algebra.</li> <li>The <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> is an alternating algebra.</li> <li>The <a href="/facts/Cohomology_ring/eyUR2o1d">cohomology ring</a> of a <a href="/facts/Topological_space/v2ut7CbK">topological space</a> is an alternating algebra.</li></ul> <h2 id="properties">Properties</h2> <ul><li>The algebra formed as the <a href="/facts/Direct_sum_of_modules/Y6PBJnjg">direct sum</a> of the homogeneous subspaces of even degree of an anticommutative algebra <i>A</i> is a <a href="/facts/Subalgebra/9PEHNF5r">subalgebra</a> contained in the <a href="/facts/Center_(ring_theory)/kJcejgkj">centre</a> of <i>A</i>, and is thus <a href="/facts/Associative_algebra/DRfoF6KV">commutative</a>.</li> <li>An anticommutative algebra <i>A</i> over a (commutative) base <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> <i>R</i> in which 2 is not a <a href="/facts/Zero_divisor/5GraAU8o">zero divisor</a> is alternating.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Alternating_multilinear_map/ssrxVffJ">Alternating multilinear map</a></li> <li><a href="/facts/Exterior_algebra/rdN4TvpR">Exterior algebra</a></li> <li><a href="/facts/Graded-symmetric_algebra/IJB2U8Nk">Graded-symmetric algebra</a></li> <li><a href="/facts/Supercommutative_algebra/crvGwsXC">Supercommutative algebra</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 482. <a href="/wiki/Springer_Science%2BBusiness_Media" target="_blank">/wiki/Springer_Science%2BBusiness_Media</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Nicolas Bourbaki (1998). Algebra I. Springer Science+Business Media. p. 482. <a href="/wiki/Springer_Science%2BBusiness_Media" target="_blank">/wiki/Springer_Science%2BBusiness_Media</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> </ol>