What is the difference between a ring, a group, and a field?

fatulant · 2006-01-25 09:08:53

Anybody?

Ricky · 2006-01-25 17:19:06

http://en.wikipedia.org/wiki/Ring_%28mathematics%29

http://en.wikipedia.org/wiki/Group_%28mathematics%29

http://en.wikipedia.org/wiki/Field_%28mathematics%29

fatulant · 2006-01-30 01:33:52

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 · 2006-01-30 03:18:34

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 a-1. 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 (2006-01-30 03:21:30)

darthradius · 2006-01-30 18:41:19

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!?

fatulant · 2006-02-04 06:53:25

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 · 2006-02-04 15:38:33

does that mean to say that poor RauLiTo here typed all that 4 no reason

Ricky · 2006-02-05 04:48:16

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?

Agnishom · 2013-07-24 05:07:18

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 a-1. 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!

Bob · 2013-07-24 05:23:24

hi

The reals form a field.

https://en.wikipedia.org/wiki/Real_number

Bob

Math Is Fun Forum

#1 2006-01-25 09:08:53

What is the difference between a ring, a group, and a field?

#2 2006-01-25 17:19:06

Re: What is the difference between a ring, a group, and a field?

#3 2006-01-30 01:33:52

Re: What is the difference between a ring, a group, and a field?

#4 2006-01-30 03:18:34

Re: What is the difference between a ring, a group, and a field?

#5 2006-01-30 18:41:19

Re: What is the difference between a ring, a group, and a field?

#6 2006-02-04 06:53:25

Re: What is the difference between a ring, a group, and a field?

#7 2006-02-04 15:38:33

Re: What is the difference between a ring, a group, and a field?

#8 2006-02-05 04:48:16

Re: What is the difference between a ring, a group, and a field?

#9 2013-07-24 05:07:18

Re: What is the difference between a ring, a group, and a field?

#10 2013-07-24 05:23:24

Re: What is the difference between a ring, a group, and a field?

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