
What is the difference between a ring, a group, and a field?
 Ricky
 Moderator
Re: What is the difference between a ring, a group, and a field?
http://en.wikipedia.org/wiki/Ring_%28mathematics%29
http://en.wikipedia.org/wiki/Group_%28mathematics%29
http://en.wikipedia.org/wiki/Field_%28mathematics%29
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
Re: What is the difference between a ring, a group, and a field?
Sorry, I should have been more specific.
Can anyone explain the difference between a ring, a group, and a field in a way so that your average 13 year old can understand?
 RauLiTo
 Full Member
Re: What is the difference between a ring, a group, and a field?
the field is in the ring and the ring is in the group i don't know how to write the mathematic definition by my own because i just understand it ... it's not a biology to save it ... well ... i copied these definition from a site i hope it's useful for you
Group: Group is the most fundamental and pervasive notion of the Higher or Abstract Algebra. It's a set along with a single operation defined on its elements. The group is called additive if the symbol for the operation is "+". It's multiplicative if the symbol "·" of multiplication is used instead. But any other symbol can be used as well. There is always a unique element (1, for multiplicative, and 0, for additive, groups) that leaves elements invariant (unchanged) under the defined operation, like a+0=a. Also, for every element a there exists a unique inverse b such that, for example, in the case of the additive symbol, a+b=0 and b+a=0. Most often, however, the inverse is denoted as a1. Lastly, the group operation must be associative like in a·(b·c)= (a·b)·c. A group is commutative or Abelian if its operation is symmetric, like in a+b=b+a.
Ring: A ring is an additive commutative group in which a second operation (normally considered as multiplication) is also defined. The multiplication must be associative, i.e. a+(b+c)= (a+b)+c and the distributive law a(b + c) = ab + ac and (b + c)a = ba + ca must hold. If a ring is also a commutative multiplicative group (of course, with 0 removed) then it's called a field.
Field: A field is a ring in which multiplication is a group operation. In France (and sometimes elsewhere in Europe), the multiplicative group need not be commutative. In the US and Russia it must be.
Last edited by RauLiTo (20060131 02:21:30)
ImPo$$!BLe = NoTH!nG Go DowN DeeP iNTo aNyTHinG U WiLL FinD MaTHeMaTiCs ...
 darthradius
 Full Member
Re: What is the difference between a ring, a group, and a field?
I don't think that you stated it explicitly, (and it is probably sort of obvious) but you MUST, first and foremost, have closure under the operation for any kind of group to exist....
flatulant 13 year old...did that make sense? And where are you that you are learning group theory in middle school!?
The greatest challenge to any thinker is stating the problem in a way that will allow a solution. Bertrand Russell
Re: What is the difference between a ring, a group, and a field?
Thanks! That helped a lot.
But I am studying other subjects that interest me. I already have basic differential and integral calculus down and am starting to learn abstract algebra.
 Tigeree
 Legendary Member
Re: What is the difference between a ring, a group, and a field?
does that mean to say that poor RauLiTo here typed all that 4 no reason
People don't notice whether it's winter or summer when they're happy. ~ Anton Chekhov Cheer up, emo kid.
 Ricky
 Moderator
Re: What is the difference between a ring, a group, and a field?
I already have basic differential and integral calculus down and am starting to learn abstract algebra.
I'm in the same boat. Kind of weird how after doing a few years of calculus you go back to proving why 1 * 1 = 1, isn't it?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
 Agnishom
 Real Member

Re: What is the difference between a ring, a group, and a field?
RauLiTo wrote:the field is in the ring and the ring is in the group i don't know how to write the mathematic definition by my own because i just understand it ... it's not a biology to save it ... well ... i copied these definition from a site i hope it's useful for you
Group: Group is the most fundamental and pervasive notion of the Higher or Abstract Algebra. It's a set along with a single operation defined on its elements. The group is called additive if the symbol for the operation is "+". It's multiplicative if the symbol "·" of multiplication is used instead. But any other symbol can be used as well. There is always a unique element (1, for multiplicative, and 0, for additive, groups) that leaves elements invariant (unchanged) under the defined operation, like a+0=a. Also, for every element a there exists a unique inverse b such that, for example, in the case of the additive symbol, a+b=0 and b+a=0. Most often, however, the inverse is denoted as a1. Lastly, the group operation must be associative like in a·(b·c)= (a·b)·c. A group is commutative or Abelian if its operation is symmetric, like in a+b=b+a.
Ring: A ring is an additive commutative group in which a second operation (normally considered as multiplication) is also defined. The multiplication must be associative, i.e. a+(b+c)= (a+b)+c and the distributive law a(b + c) = ab + ac and (b + c)a = ba + ca must hold. If a ring is also a commutative multiplicative group (of course, with 0 removed) then it's called a field.
Field: A field is a ring in which multiplication is a group operation. In France (and sometimes elsewhere in Europe), the multiplicative group need not be commutative. In the US and Russia it must be.
Example please!
'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.' 'God exists because Mathematics is consistent, and the devil exists because we cannot prove it' 'Who are you to judge everything?' Alokananda
 bob bundy
 Moderator
Re: What is the difference between a ring, a group, and a field?
You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei
