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In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it.
The trivial operation x ∗ y = x (that is, the result is the first argument, no matter what the second argument is) is associative but not commutative. Likewise, the trivial operation x ∘ y = y (that is, the result is the second argument, no matter what the first argument is) is associative but not commutative.
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. [1]
The cubic plane curve (red) defined by the equation y 2 = x 2 (x + 1) is singular at the origin, i.e., the ring k[x, y] / y 2 − x 2 (x + 1), is not a regular ring. The tangent cone (blue) is a union of two lines, which also reflects the singularity. The k-vector space m/m 2 is an algebraic incarnation of the cotangent space.
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality (+) = + is always true in elementary algebra.