Lucifer

Definition

This construction can be done in the following way: if I is a Ensemble of indices and A_i is a ring for any index I of I , then the Cartesian Produit Π_{I in I} has _I can be provided with a structure of ring by defining the operations component by component, i.e

( a_i ) + ( b_i ) = ( a_i + b_i )

( a_i ) · ( b_i ) = ( a_i · b_i ).

In the place of Π_{1≤i≤ K} A_i we can also write A₁ × A₂ ×… × A_k .

Examples

The most important example is the Anneau Z/nZ entireties modulo N (see modular Arithmétique). If N is written like a product of powers of factors first (see the fundamental Théorème of arithmetic the):

N = p ₁^{N ₁} p ₂^{N ₂}… p _k^{N _K}

where the all p_i are distinct, then

\ mathbb Z/n \ mathbb Z

is naturally isomorphous with the ring produced

\ mathbb Z {p_1} ^ {n_1} \ mathbb Z \ times \ mathbb Z {p_2} ^ {n_2} \ mathbb Z \ times \ cdots \ mathbb Z {p_k} ^ {n_k} \ mathbb Z

That rises from the Théorème of the Chinese remainders.

Properties

If has = Π_{I in I} has _I is a product of rings, then for all I in I we have a surjective Homomorphisme p_i : has → A_i which projects an element of the product on the component I ème. The product has , as well as projections p_i , have the universal Propriété following:

if B is an unspecified ring and f_i : B → A_i is a morphism of rings for all I in I , then it exists a single morphism of rings F : B → has such as for all I in I , p_i O F = f_i .

If I is a Idéal (on the left, on the right or on the two sides) of has , then there exist ideals (on the left, on the right or on the two sides respectively) I_i of A_i such as I = Π_{I in I} I_i . Conversely, such a product of ideals is an ideal of has . I is a Idéal first of has if and only if all the I_i except one are equal to A_i and the remainder I_i is an ideal first of A_i .

An element X of has is invertible if and only if all its components are invertible, i.e if and only if p_i ( X ) is an invertible element of A_i for all I of T . The group of the invertible elements of has is the produced of the groups of invertible of A_i .

A product of more than one ring not no one always has dividing of zero: if X is an element of the product whose components are null except p_i ( X ), and there is an element of the product of which all the components are null except p_j ( there ) (with I ≠ J ), then xy = 0 in the produced ring.

See too

Produces groups Topology produced

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