Is a Field a Commutative Ring with Nonzero Unity?

[ehrenfest]

[SOLVED] field theory

Homework Statement

Is the following sentence true:

A field is a ring with nonzero unity such that the set of nonzero elements of F is a group under multiplication.

?

Homework Equations

The Attempt at a Solution

I think it is false. To make it correct, we must require that the ring be commutative.
Am I correct?

 

[masnevets]

If you're asking whether that definition implies commutativity, then you're correct, it does not. For example, you can take the quaternions with rational coefficients. They form a noncommutative field.

If you're asking whether the definition of a field requires commutativity, it depends. Most people would define it that way, but some texts (for example, Serre's books) do not. Normally, one calls a noncommutative field a skew field or a division ring.

 

[ehrenfest]

Here is the definition of a field in my book (Farleigh):

"Let R be a ring with unity 1 not equal to 0. An element u in R is a unit of R if it has a multiplicative inverse in R. If every nonzero element of R is a unit, then R is a division ring. A field is a commutative division ring."

Please confirm the truth of the following statements:

1)The nonzero elements of a field form a group under the multiplication in the field

2)A commutative ring with nonzero unity such that the set of nonzero elements of F is a group under multiplication is also a field.

 

[masnevets]

Here is the definition of a field in my book (Farleigh):

"Let R be a ring with unity 1 not equal to 0. An element u in R is a unit of R if it has a multiplicative inverse in R. If every nonzero element of R is a unit, then R is a division ring. A field is a commutative division ring."

Please confirm the truth of the following statements:

1)The nonzero elements of a field form a group under the multiplication in the field

2)A commutative ring with nonzero unity such that the set of nonzero elements of F is a group under multiplication is also a field.

1) yes
2) yes