= Noncommutative algebra
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<Noncommutative algebra> studies <rings> and <algebras> without assuming $ab=ba$. <Matrix algebras>, <group algebras>, and <Weyl algebras> are central examples. The order of multiplication becomes part of the mathematical data.
For instance, a <left ideal> is closed under multiplication on the left, while a <right ideal> is closed under multiplication on the right; only a <two-sided ideal> supports a <quotient ring>. Likewise left and right <modules> can differ. Studying their categories through <Morita equivalence> often reveals similarities between <rings> that are not themselves isomorphic.
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