³1username³2³2name³2text_default³2version³21³2ts³2980230443³2ip³2216.165.4.144³2data³2newauthor³31³3minor³30³3text³3A Ring is a commutative groupunder an operation +,
along with an associative operation * with unit element 1,
such that multiplication distributes over addition.
³3summary³3*³2tscreate³2980230443³2keepts³2980233153³2id³21088³2host³2DIALN-ASYNC389.DIAL.NET.NYU.EDU³2revision³21³1username³2LarrySanger³2name³2text_default³2version³21³2ts³2980233153³2ip³224.234.246.58³2data³2text³3A Ring is a commutative group under an operation +,
along with an associative operation * with unit element 1,
such that multiplication distributes over addition.
----
Why not define some of the simpler terminology here and used in other mathematical articles?  Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger
³3minor³30³3newauthor³31³3summary³3*³2tscreate³2980230443³2keepts³2980233381³2id³21001³2host³2dhcp058.246.lvcm.com³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980233381³2ip³2129.97.130.40³2data³2newauthor³31³3minor³30³3text³3A Ring is a commutative group under an operation +, together with a second operation * s.t.

1.  a*(b*c) = (a*b)*c
1.  a*(b+c) = (a*b)+(a*c), (a+b)*c = (a*c)+(b*c)

Very often the definition of a ring is taken to require a multiplicative identity, or unity, denoted 1.  Nearly all important rings actually satisfy this.  It has the disadvantage, however, of making ring ideals not subrings, as compared with their group-analog, the normal subgroups.  It has the advantage of adding a lot more structure to 

A justification for the definition of a ring is that collections of commutative group homomorphisms closed under composition form rings (much the same way bijections form groups).  The commutative group is referred to as a module under the ring; the module is called faithful if the actions of ring elements are each different.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain.  The most interesting of these are FielDs, where every non-zero element has a multiplicative inverse.

Some important rings include the integers, modular arithmetics, and polynomials.
----
Why not define some of the simpler terminology here and used in other mathematical articles?  Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) -JG
³3summary³3³2tscreate³2980230443³2keepts³2980233738³2id³21076³2host³2cnts3p40.uwaterloo.ca³2revision³23³1username³2LarrySanger³2name³2text_default³2version³21³2ts³2980233738³2ip³224.234.246.58³2data³2text³3A Ring is a commutative group under an operation +, together with a second operation * s.t.

1.  a*(b*c) = (a*b)*c
1.  a*(b+c) = (a*b)+(a*c), (a+b)*c = (a*c)+(b*c)

Very often the definition of a ring is taken to require a multiplicative identity, or unity, denoted 1.  Nearly all important rings actually satisfy this.  It has the disadvantage, however, of making ring ideals not subrings, as compared with their group-analog, the normal subgroups.  It has the advantage of adding a lot more structure to 

A justification for the definition of a ring is that collections of commutative group homomorphisms closed under composition form rings (much the same way bijections form groups).  The commutative group is referred to as a module under the ring; the module is called faithful if the actions of ring elements are each different.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain.  The most interesting of these are FielDs, where every non-zero element has a multiplicative inverse.

Some important rings include the integers, modular arithmetics, and polynomials.
----
Why not define some of the simpler terminology here and used in other mathematical articles?  Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) -JG

Except for the terms you use to define 'ring'; those are more primitive than 'ring', I hope! -- LarrySanger
³3minor³31³3newauthor³31³3summary³3*³2tscreate³2980230443³2keepts³2980237571³2id³21001³2host³2dhcp058.246.lvcm.com³2revision³24³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980237571³2ip³2129.97.182.28³2data³2newauthor³31³3minor³31³3text³3A Ring is a commutative group under an operation +, together with a second operation * s.t.

* a*(b*c) = (a*b)*c
* a*(b+c) = (a*b)+(a*c), (a+b)*c = (a*c)+(b*c)

Very often the definition of a ring is taken to require a multiplicative identity, or unity, denoted 1.  Nearly all important rings actually satisfy this.  It has the disadvantage, however, of making ring ideals not subrings, as compared with their group-analog, the normal subgroups.  It has the advantage of adding a lot more structure to 

Every ring can be represented in terms of actions on MathematicalGroups.  The two structures together form what are known as ModulEs.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain.  The most interesting of these are FielDs, where every non-zero element has a multiplicative inverse.

Some important rings include the integers, modular arithmetics, and polynomials.
----
Why not define some of the simpler terminology here and used in other mathematical articles?  Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) -JG

Except for the terms you use to define 'ring'; those are more primitive than 'ring', I hope! -- LarrySanger
³3summary³3*³2tscreate³2980230443³2keepts³2980239708³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³25³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980239708³2ip³2129.97.182.28³2data³2text³3A Ring is a commutative group under an operation +, together with a second operation * s.t.

* a*(b*c) = (a*b)*c
* a*(b+c) = (a*b)+(a*c), (a+b)*c = (a*c)+(b*c)

Very often the definition of a ring is taken to require a multiplicative identity, or unity, denoted 1.  Nearly all important rings actually satisfy this.  It has the disadvantage, however, of making ring ideals not subrings, as compared with their group-analog, the normal subgroups.  It has the advantage of adding a lot more structure to 

Sets of commutative group HomoMorphisms form rings under addition and composition, provided they are closed under all necessary operations.  An isomorphism from a ring to such a collection is called a representation of the ring, and groups under ring-actions are referred to as ModulEs.  Every ring has some sort of faithful representation.

A subring (not necessarily with identity) closed under multiplication by ''arbitrary ring'' elements is called an ideal.  These relate to the kernels of ring HomoMorphisms the same way NormalSubGroups relate to those of GroupHomomorphisms.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain.  In such rings, multiplicative cancellation is possible.  Of particular interest are FielDs, IntegralDomain''''s where every non-zero element has a muliplicative inverse.

Every ring contains a unique smallest subring (with identity), isomorphic to either the IntegerNumbers or a ModularArithmetic.  ModularArithmetics of prime order are actually fields, but those of composite order aren't even integral domains.  Every field containing the IntegerNumbers contains a unique smallest field isomorphic to the RationalNumbers.
----
Why not define some of the simpler terminology here and used in other mathematical articles?  Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) -JG

Except for the terms you use to define 'ring'; those are more primitive than 'ring', I hope! -- LarrySanger
³3minor³30³3newauthor³30³3summary³3³2tscreate³2980230443³2keepts³2980240657³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³26³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980240657³2ip³2129.97.182.28³2data³2newauthor³30³3minor³31³3text³3A Ring is a commutative group under an operation +, together with a second operation * s.t.

* a*(b*c) = (a*b)*c
* a*(b+c) = (a*b)+(a*c), (a+b)*c = (a*c)+(b*c)

Very often the definition of a ring is taken to require a multiplicative identity, or unity, denoted 1.  Nearly all important rings actually satisfy this.  It has the disadvantage, however, of making ring ideals not subrings, as compared with their group-analog, the normal subgroups.  It has the advantage of adding a lot more structure to 

Sets of commutative group HomoMorphisms form rings under addition and composition, provided they are closed under all necessary operations.  An isomorphism from a ring to such a collection is called a representation of the ring, and groups under ring-actions are referred to as ModulEs.  Every ring has some sort of faithful representation.

A subring (not necessarily with identity) closed under multiplication by ''arbitrary ring'' elements is called an ideal.  These relate to the kernels of ring HomoMorphisms the same way NormalSubGroups relate to those of GroupHomomorphisms.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain.  In such rings, multiplicative cancellation is possible.  Of particular interest are FielDs, IntegralDomain''''s where every non-zero element has a muliplicative inverse.

Every ring contains a unique smallest subring (with identity), isomorphic to either the IntegerNumbers or a ModularArithmetic.  Every field containing the IntegerNumbers contains a unique smallest field isomorphic to the RationalNumbers.
----
Why not define some of the simpler terminology here and used in other mathematical articles?  Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) -JG

Except for the terms you use to define 'ring'; those are more primitive than 'ring', I hope! -- LarrySanger
³3summary³3*³2tscreate³2980230443³2keepts³2980241423³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³27³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980241423³2ip³2129.97.182.28³2data³2text³3A Ring is a commutative group under an operation +, together with a second operation * s.t.

* a*(b*c) = (a*b)*c
* a*(b+c) = (a*b)+(a*c), (a+b)*c = (a*c)+(b*c)

Very often the definition of a ring is taken to require a multiplicative identity, or unity, denoted 1.  Nearly all important rings actually satisfy this.  It has the disadvantage, however, of making ring ideals not subrings, as compared with their group-analog, the normal subgroups.  It has the advantage of adding a lot more structure to 

Sets of commutative group HomoMorphisms form rings under addition and composition, provided they are closed under all necessary operations.  An isomorphism from a ring to such a collection is called a representation of the ring, and groups under ring-actions are referred to as ModulEs.  Every ring has some sort of faithful representation.

A subring (not necessarily with identity) closed under multiplication by ''arbitrary ring'' elements is called an ideal.  These relate to the kernels of ring HomoMorphisms the same way NormalSubGroups relate to those of GroupHomomorphisms.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain.  In such rings, multiplicative cancellation is possible.  Of particular interest are FielDs, IntegralDomain''''s where every non-zero element has a muliplicative inverse.

Every ring contains a unique smallest subring (with identity), isomorphic to either the IntegerNumbers or a ModularArithmetic.  Every field containing the IntegerNumbers contains a unique smallest field isomorphic to the RationalNumbers.
----
Why not define some of the simpler terminology here and used in other mathematical articles?  Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) -JG

Except for the terms you use to define 'ring'; those are more primitive than 'ring', I hope! -- LarrySanger

Fair enough.  I think I'm going to shoot for breadth first, then depth, though.  Once there's pages enough to at least be able to properly define the RealNumbers, I'll go back and sort things out.  Sound 'K?
³3minor³31³3newauthor³30³3summary³3³2tscreate³2980230443³2keepts³2980496897³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³28³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980496897³2ip³2129.97.130.42³2data³2newauthor³30³3minor³31³3text³3deleted in favor of RinG
³3summary³3*³2tscreate³2980230443³2keepts³2980895031³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³29