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The Egyptians used the commutative property of multiplication to simplify computing products. [7] [8] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of ...
The algebraic structure (, +) is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group. The smallest group containing the natural numbers is the integers.
Every vector space is a free module, [1] but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S.
The free commutative semigroup is the subset of the free commutative monoid that contains all multisets with elements drawn from A except the empty multiset. The free partially commutative monoid , or trace monoid , is a generalization that encompasses both the free and free commutative monoids as instances.
The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a. Next we will prove the base case b = 1, that 1 commutes with everything, i.e. for all natural numbers a, we have a + 1 = 1 + a.
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; [1] where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit element. [2]
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