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  • In general, two elements in a commutative ring can have no least common multiple or more than one.
  • In the non-commutative ring case the same definition does not always work.
  • The real numbers can be extended to a wheel, as can any commutative ring.
  • The center of a division ring is commutative and therefore a field.
  • Similarly, union is commutative, so the sets can be written in any order.
  • The first three relations show that products of the (non-real) basis elements are anti-commutative.
  • He worked in many areas of algebra, mainly in non-commutative ring theory.
  • We aim to show that the addition of natural numbers is commutative.
  • A commutative ring is left primitive if and only if it is a field.
  • Records of the implicit use of the commutative property go back to ancient times.
  • Today the commutative property is a well known and basic property used in most branches of mathematics.
  • Here, the base ring is a ring with unity, though possibly non-commutative.
  • The commutative law of addition can be used to rearrange terms into any preferred order.
  • Therefore the mean should also be observed in the same way in both distributive and commutative justice. Cited from Summa Theologica, Part II-II (Secunda Secundae), by Thomas Aquinas
  • Hence all these actions belong to the one same species of justice, namely commutative justice. Cited from Summa Theologica, Part II-II (Secunda Secundae), by Thomas Aquinas
  • Addition is commutative and associative so the order the terms are added in does not matter.
  • In contrast, the commutative property states that the order of the terms does not affect the final result.
  • A commutative ring is always equal to its opposite ring.
  • If R is a commutative ring, then the above three notions are all the same.
  • The same is true with the ring of integers replaced by any other commutative ring.
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