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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 .
A product of two or more non-trivial rings always has nonzero zero divisors: if x is an element of the product whose coordinates are all zero except p i (x) and y is an element of the product with all coordinates zero except p j (y) where i ≠ j, then xy = 0 in the product ring.
Examples of limits and colimits in Ring include: The ring of integers Z is an initial object in Ring. The zero ring is a terminal object in Ring. The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple.
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.
In set theory, a Cartesian product is a mathematical operation which returns a set (or product set) from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) —where a ∈ A and b ∈ B. [5] The class of all things (of a given type) that have Cartesian products is called a Cartesian ...
If A and B are sets, then the Cartesian product (or simply product) is defined to be: A × B = {(a,b) | a ∈ A and b ∈ B}. That is, A × B is the set of all ordered pairs whose first coordinate is an element of A and whose second coordinate is an element of B.
Two prominent examples occur for Banach spaces and Hilbert spaces. In some classical texts, the phrase "direct sum of algebras over a field" is also introduced for denoting the algebraic structure that is presently more commonly called a direct product of algebras; that is, the Cartesian product of the underlying sets with the componentwise ...