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Encyclopedia > Direct product (ring theory)

In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e. Euclid, detail from The School of Athens by Raphael. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, an index set is another name for a function domain. ... In mathematics, the Cartesian product (or direct product) of two sets X and Y, denoted X × Y, is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y: The Cartesian product is named after René Descartes...

(ai) + (bi) = (ai + bi)
(ai) · (bi) = (ai · bi)

The resulting ring is called a direct product of the rings Ri. The direct product of finitely many rings R1,...,Rk is also written as R1 × R2 × ... × Rk.


Examples

The most important example is the ring Z/nZ of integers modulo n. If n is written as a product of prime powers (see fundamental theorem of arithmetic): The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ...

n={p_1}^{n_1} {p_2}^{n_2} cdots {p_k}^{n_k}

where the pi are distinct primes, then Z/nZ is naturally isomorphic to the product ring In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...

mathbf{Z}/{p_1}^{n_1}mathbf{Z}  times  mathbf{Z}/{p_2}^{n_2}mathbf{Z}  times  cdots  times  mathbf{Z}/{p_k}^{n_k}mathbf{Z}

This follows from the Chinese remainder theorem. link titleThe Chinese remainder theorem (CRT) is the name for several related results in abstract algebra and number theory. ...


Properties

If R = Πi in I Ri is a product of rings, then for every i in I we have a surjective ring homomorphism pi : R -> Ri which projects the product on the i-th coordinate. The product R, together with the projections pi, has the following universal property: In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

if S is any ring and fi : SRi is a ring homomorphism for every i in I, then there exists precisely one ring homomorphism f : SR such that pi o f = fi for every i in I.

This shows that the product of rings is an instance of products in the sense of category theory. In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ...


If A is a (left, right, two-sided) ideal in R, then there exist (left, right, two-sided) ideals Ai in Ri such that A = Πi in I Ai. Conversely, every such product of ideals is an ideal in R. A is a prime ideal in R if and only if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...


An element x in R is a unit if and only if all of its components are units, i.e. if and only if pi(x) is a unit in Ri for every i in I. The group of units of R is the product of the groups of units of Ri.


A product of more than one non-zero rings always has zero divisors: if x is an element of the product all of whose coordinates are zero except pi(x), and y is an element of the product with all coordinates zero except pj(y) (with ij), then xy = 0 in the product ring. In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ...


See also


 
 

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