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Concept
Direct product
Summary
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one talks about the , which formalizes these notions.
Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance.
There is also the direct sum – in some areas this is used interchangeably, while in others it is a different concept.
If we think of as the set of real numbers, then the direct product is just the Cartesian product
If we think of as the group of real numbers under addition, then the direct product still has as its underlying set. The difference between this and the preceding example is that is now a group, and so we have to also say how to add their elements. This is done by defining
If we think of as the ring of real numbers, then the direct product again has as its underlying set. The ring structure consists of addition defined by and multiplication defined by
Although the ring is a field, is not one, because the element does not have a multiplicative inverse.
In a similar manner, we can talk about the direct product of finitely many algebraic structures, for example, This relies on the fact that the direct product is associative up to isomorphism. That is, for any algebraic structures and of the same kind. The direct product is also commutative up to isomorphism, that is, for any algebraic structures and of the same kind. We can even talk about the direct product of infinitely many algebraic structures; for example we can take the direct product of countably many copies of which we write as
Direct product of groups and Direct sum
In group theory one can define the direct product of two groups and denoted by For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by
It is defined as follows:
the set of the elements of the new group is the Cartesian product of the sets of elements of that is
on these elements put an operation, defined element-wise:
Note that may be the same as
This construction gives a new group.
Official source
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