Difference between revisions of "Product of rings"

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In [[mathematics]], it is possible to combine several [[ring (mathematics)|rings]] into one large '''product ring'''. This is done as follows: if ''I'' is some [[index set]] and ''R<sub>i</sub>'' is a ring for every ''i'' in ''I'', then the [[cartesian product]] Π<sub>''i'' in ''I''</sub> ''R''<sub>''i''</sub> can be turned into a ring by defining the operations coordinatewise, i.e.
:(''a<sub>i</sub>'') + (''b<sub>i</sub>'') = (''a<sub>i</sub>'' + ''b<sub>i</sub>'')
:(''a<sub>i</sub>'') · (''b<sub>i</sub>'') = (''a<sub>i</sub>'' · ''b<sub>i</sub>'')
The resulting ring is called a '''direct product''' of the rings ''R''<sub>''i''</sub>. The direct product of finitely many rings ''R''<sub>1</sub>,...,''R''<sub>''k''</sub> is also written as ''R''<sub>1</sub> &times; ''R''<sub>2</sub> &times; ... &times; ''R''<sub>''k''</sub> or ''R''<sub>1</sub> ⊕ ''R''<sub>2</sub> ⊕ ... ⊕ ''R''<sub>''k''</sub>, and can also be called the '''direct sum''' (and sometimes the '''complete direct sum'''<ref>{{harvnb|Herstein|2005|loc=p. 52}}</ref>) of the rings ''R<sub>i</sub>''.
<!-- Stuff removed from [[ring (mathematics)]]:
Let ''R'' and ''S'' be rings. Then the product ''R'' X ''S'' can be equipped with the following natural ring structure:


* (''r''<sub>1</sub>, ''s''<sub>1</sub>) + (''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub> + ''r''<sub>2</sub>, ''s''<sub>1</sub> + ''s''<sub>2</sub>)
In [[mathematics]], it is possible to combine several [[ring (mathematics)|rings]] into one large '''product ring'''. This is done as follows: if ''I'' is some [[index set]] and ''R<sub>i</sub>'' is a ring for every ''i'' in ''I'', then the [[cartesian product]] {{nowrap|Π<sub>''i'' ''I''</sub> ''R''<sub>''i''</sub>}} can be turned into a ring by defining the operations coordinate-wise.


* (''r''<sub>1</sub>, ''s''<sub>1</sub>) '''·''' (''r''<sub>2</sub>, ''s''<sub>2</sub>) = (''r''<sub>1</sub> '''·''' ''r''<sub>2</sub>, ''s''<sub>1</sub> '''·''' ''s''<sub>2</sub>)
The resulting ring is called a '''direct product''' of the rings ''R''<sub>''i''</sub>. The direct product of finitely many rings coincides with the [[direct sum]] of rings.
 
for every ''r''<sub>1</sub>, ''r''<sub>2</sub> in ''R'' and ''s''<sub>1</sub>, ''s''<sub>2</sub> in ''S''. The ring ''R'' X ''S'' with the above operations of addition and operation (derived from ''R'' and ''S'') is called the '''[[Direct product of rings|direct product]]''' of ''R'' with ''S''. By repeating this process ''n'' times with ''n'' rings, one can form what is known as the '''''n''-fold product''' of rings. The product is natural because the [[quotient ring]] ''R'' X ''S'' / ''R'' is isomorphic to ''S'' and similarly ''R'' X ''S'' / ''S'' is isomorphic to ''R''.
-->


==Examples==
==Examples==
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==Properties==
==Properties==
If ''R'' = Π<sub>''i'' in ''I''</sub> ''R''<sub>''i''</sub> is a product of rings, then for every ''i'' in ''I'' we have a [[surjective]] [[ring homomorphism]] ''p<sub>i</sub>'': ''R'' → ''R<sub>i</sub>'' which projects the product on the ''i''-th coordinate. The product ''R'', together with the projections ''p<sub>i</sub>'', has the following [[universal property]]:  
If {{nowrap|1=''R'' = Π<sub>''i'' ''I''</sub> ''R''<sub>''i''</sub>}} is a product of rings, then for every ''i'' in ''I'' we have a [[surjective]] [[ring homomorphism]] {{nowrap|''p<sub>i</sub>'': ''R'' → ''R<sub>i</sub>''}} which projects the product on the ''i''th coordinate. The product ''R'', together with the projections ''p<sub>i</sub>'', has the following [[universal property]]:  


:if ''S'' is any ring and ''f<sub>i</sub>'': ''S'' &rarr; ''R<sub>i</sub>'' is a ring homomorphism for every ''i'' in ''I'', then there exists ''precisely one'' ring homomorphism ''f'': ''S'' &rarr; ''R'' such that ''p<sub>i</sub>'' o ''f'' = ''f<sub>i</sub>'' for every ''i'' in ''I''.
:if ''S'' is any ring and {{nowrap|''f<sub>i</sub>'': ''S'' ''R<sub>i</sub>''}} is a ring homomorphism for every ''i'' in ''I'', then there exists ''precisely one'' ring homomorphism {{nowrap|''f'': ''S'' ''R''}} such that {{nowrap|1=''p<sub>i</sub>'' ''f'' = ''f<sub>i</sub>''}} for every ''i'' in ''I''.


This shows that the product of rings is an instance of [[product (category theory)|products in the sense of category theory]]. However, despite also being called the direct sum of rings when ''I'' is finite, the product of rings is not a [[coproduct]] in the sense of category theory. In particular, if ''I'' has more than one element, the inclusion map ''R<sub>i''</sub>&nbsp;&nbsp;''R'' is not ring homomorphism as it does not map the identity in ''R<sub>i''</sub> to the identity in ''R''.
This shows that the product of rings is an instance of [[product (category theory)|products in the sense of category theory]]. However, despite also being called the direct sum of rings when ''I'' is finite, the product of rings is not a [[coproduct]] in the sense of category theory. In particular, if ''I'' has more than one element, the inclusion map {{nowrap|''R<sub>i''</sub> → ''R''}} is not ring homomorphism as it does not map the identity in ''R<sub>i''</sub> to the identity in ''R''.


If ''A<sub>i</sub>'' in ''R<sub>i</sub>'' is an [[ideal (ring theory)|ideal]] for each ''i'' in ''I'', then ''A'' = Π<sub>''i'' in ''I''</sub> ''A<sub>i</sub>'' is an ideal of ''R''.  If ''I'' is finite, then the converse is true, i.e. every ideal of ''R'' is of this form.  However, if ''I'' is infinite and the rings ''R<sub>i</sub>'' are non-zero, then the converse is false; the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the ''R<sub>i</sub>''.  The ideal ''A'' is a [[prime ideal]] in ''R'' if all but one of the ''A<sub>i</sub>'' are equal to ''R<sub>i</sub>'' and the remaining ''A<sub>i</sub>'' is a prime ideal in ''R<sub>i</sub>''. However, the converse is not true when ''I'' is infinite. For example, the [[Direct sum of modules|direct sum]] of the ''R<sub>i</sub>'' form an ideal not contained in any such ''A'', but the [[axiom of choice]] gives that it is contained in some [[maximal ideal]] which is [[a fortiori]] prime.
If ''A<sub>i</sub>'' in ''R<sub>i</sub>'' is an [[ideal (ring theory)|ideal]] for each ''i'' in ''I'', then {{nowrap|1=''A'' = Π<sub>''i'' ''I''</sub> ''A<sub>i</sub>''}} is an ideal of ''R''.  If ''I'' is finite, then the converse is true, i.e. every ideal of ''R'' is of this form.  However, if ''I'' is infinite and the rings ''R<sub>i</sub>'' are non-zero, then the converse is false; the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the ''R<sub>i</sub>''.  The ideal ''A'' is a [[prime ideal]] in ''R'' if all but one of the ''A<sub>i</sub>'' are equal to ''R<sub>i</sub>'' and the remaining ''A<sub>i</sub>'' is a prime ideal in ''R<sub>i</sub>''. However, the converse is not true when ''I'' is infinite. For example, the [[Direct sum of modules|direct sum]] of the ''R<sub>i</sub>'' form an ideal not contained in any such ''A'', but the [[axiom of choice]] gives that it is contained in some [[maximal ideal]] which is [[a fortiori]] prime.


An element ''x'' in ''R'' is a unit if and only if all of its components are units, i.e. if and only if ''p<sub>i</sub>''(''x'') is a unit in ''R<sub>i</sub>'' for every ''i'' in ''I''. The group of units of ''R'' is the [[direct product of groups|product]] of the groups of units of ''R<sub>i</sub>''.
An element ''x'' in ''R'' is a unit if and only if all of its components are units, i.e. if and only if {{nowrap|''p<sub>i</sub>''(''x'')}} is a unit in ''R<sub>i</sub>'' for every ''i'' in ''I''. The group of units of ''R'' is the [[direct product of groups|product]] of the groups of units of ''R<sub>i</sub>''.


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 ''p<sub>i</sub>''(''x''), and ''y'' is an element of the product with all coordinates zero except ''p<sub>j</sub>''(''y'') (with ''i'' ≠ ''j''), then ''xy'' = 0 in the product ring.
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 {{nowrap|''p<sub>i</sub>''(''x'')}}, and ''y'' is an element of the product with all coordinates zero except {{nowrap|''p<sub>j</sub>''(''y'')}} (with {{nowrap|''i'' ≠ ''j''}}), then {{nowrap|1=''xy'' = 0}} in the product ring.


==See also==
==See also==
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[[Category:Ring theory]]
[[Category:Ring theory]]
[[Category:Binary operations]]
[[Category:Binary operations]]
[[fr:Produit d'anneaux]]
[[nl:Product van ringen]]

Latest revision as of 02:06, 4 August 2013

{{ safesubst:#invoke:Unsubst||$N=Refimprove |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }}

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 ΠiI Ri can be turned into a ring by defining the operations coordinate-wise.

The resulting ring is called a direct product of the rings Ri. The direct product of finitely many rings coincides with the direct sum of rings.

Examples

An 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):

where the pi are distinct primes, then Z/nZ is naturally isomorphic to the product ring

This follows from the Chinese remainder theorem.

Properties

If R = ΠiI Ri is a product of rings, then for every i in I we have a surjective ring homomorphism pi: RRi which projects the product on the ith coordinate. The product R, together with the projections pi, has the following universal property:

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 pif = fi for every i in I.

This shows that the product of rings is an instance of products in the sense of category theory. However, despite also being called the direct sum of rings when I is finite, the product of rings is not a coproduct in the sense of category theory. In particular, if I has more than one element, the inclusion map RiR is not ring homomorphism as it does not map the identity in Ri to the identity in R.

If Ai in Ri is an ideal for each i in I, then A = ΠiI Ai is an ideal of R. If I is finite, then the converse is true, i.e. every ideal of R is of this form. However, if I is infinite and the rings Ri are non-zero, then the converse is false; the set of elements with all but finitely many nonzero coordinates forms an ideal which is not a direct product of ideals of the Ri. The ideal A is a prime ideal in R if all but one of the Ai are equal to Ri and the remaining Ai is a prime ideal in Ri. However, the converse is not true when I is infinite. For example, the direct sum of the Ri form an ideal not contained in any such A, but the axiom of choice gives that it is contained in some maximal ideal which is a fortiori prime.

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.

See also

Notes

References

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