A least common multiple of a and b is a common multiple that is minimal in the sense that for any other common multiple n of a and b, m divides n.In general, two elements in a commutative ring can have no least common multiple or more than one.However, any two least common multiples of the same pair of elements are associates.…

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring.In the non-commutative ring case the same definition does not always work.This has resulted in several radicals generalizing the commutative case in distinct ways.…

The real numbers can be extended to a wheel, as can any commutative ring.

The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same, and define the rank of a matrix.The center of a division ring is commutative and therefore a field.Every division ring is therefore a division algebra over its center.…

Similarly, union is commutative, so the sets can be written in any order.The empty set is an identity element for the operation of union.…

The first three relations show that products of the (non-real) basis elements are anti-commutative.

In all, Cohn wrote nearly 200 mathematical papers.He worked in many areas of algebra, mainly in non-commutative ring theory.His first papers, covering many topics, were published in 1952.…

Special rules, called wave rules, can then be used in a terminating fashion to manipulate the target expression until it the source expression can be used to complete the proof.We aim to show that the addition of natural numbers is commutative.This is an elementary property, and the proof is by routine induction.…

A commutative ring is left primitive if and only if it is a field.Being left primitive is a Morita invariant property.…

Records of the implicit use of the commutative property go back to ancient times.The Egyptians used the commutative property of multiplication to simplify computing products.…

Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions.Today the commutative property is a well known and basic property used in most branches of mathematics.…

A module E is called the injective hull of a module M, if E is an essential extension of M, and E is injective.Here, the base ring is a ring with unity, though possibly non-commutative.In some cases, for R a subring of a self-injective ring S, the injective hull of R will also have a ring structure.…

The commutative law of addition can be used to rearrange terms into any preferred order.

Now the mean is taken in the same way in all the parts of temperance or fortitude.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

In all these actions, whether voluntary or involuntary, the mean is taken in the same way according to the equality of repayment.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

Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number 1 is the most basic form of counting.Addition is commutative and associative so the order the terms are added in does not matter.The identity element of addition (the additive identity) is 0, that is, adding 0 to any number yields that same number.…

The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms doesn't change.In contrast, the commutative property states that the order of the terms does not affect the final result.Most commutative operations encountered in practice are also associative.…

A ring and its opposite ring are anti-isomorphic.A commutative ring is always equal to its opposite ring.A non-commutative ring may or may not be isomorphic to its opposite ring.…

A principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R.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.These statements follow from analogous statements for the elementary symmetric polynomials, due to the indicated possibility of expressing either kind of symmetric polynomials in terms of the other kind.…