³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980496870³2ip³2129.97.130.42³2data³2newauthor³31³3minor³31³3text³3A 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 HomoMorphism''s'' 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 ModulE''s''.  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 HomoMorphism''s'' the same way NormalSubGroups relate to those of GroupHomomorphism''s''.

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 FielD''s'', 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.
³3summary³3*³2tscreate³2980496870³2keepts³2980697696³2id³21076³2host³2cnts3p42.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980697696³2ip³2129.97.13.16³2data³2text³3A 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.

Some important concepts: SubRing''s'' (including ideals)

Sets of commutative group HomoMorphism''s'' 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 ModulE''s''.  Every ring has some sort of faithful representation.

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 FielD''s'', IntegralDomain''s'' where every non-zero element has a muliplicative inverse.

³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980496870³2keepts³2980715107³2id³21076³2host³2cnts6p16.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980715107³2ip³2129.97.13.28³2data³2newauthor³30³3minor³31³3text³3A 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.

Some important concepts: SubRing''s'' and ideals, ModulE''s''

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 FielD''s'', IntegralDomain''s'' where every non-zero element has a muliplicative inverse.

³3summary³3Removed module stuff³2tscreate³2980496870³2keepts³2980717964³2id³21076³2host³2cnts6p28.uwaterloo.ca³2revision³23