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

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Topic review (newest first)

bob bundy
2013-07-25 03:23:24

hi

The reals form a field.

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

Bob

Agnishom
2013-07-25 03:07:18

RauLiTo wrote:

the field is in the ring and the ring is in the group cool
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 smile

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(bc)= (ab)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!

Ricky
2006-02-06 03:48:16

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?

Tigeree
2006-02-05 14:38:33

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

fatulant
2006-02-05 05:53:25

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.

darthradius
2006-01-31 17:41:19

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!?  smile

RauLiTo
2006-01-31 02:18:34

the field is in the ring and the ring is in the group cool
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 smile

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(bc)= (ab)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.

fatulant
2006-01-31 00:33:52

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?

Ricky
2006-01-26 16:19:06

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
2006-01-26 08:08:53

Anybody?

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