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roni1

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What are the differences between

It's for an article that I write.

**ring**and**field**?It's for an article that I write.

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In summary: This change allows a field to have 0 as a member, and is formally known as the axiom of extensionality. In summary, a field has more conditions/axioms than a ring. For instance, all elements in a field, except the zero-element ($0$), must have a multiplicative inverse. And multiplication in a field must be commutative. Depending on the text you have, a ring may need to have a multiplicative identity ($1$) or not. Either way, a field must have it, and it must be distinct from the zero-element ($0$).

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roni1

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What are the differences between **ring** and **field**?

It's for an article that I write.

It's for an article that I write.

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I like Serena

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MHB

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roni said:What are the differences betweenringandfield?

It's for an article that I write.

Hi roni,

A field has more conditions/axioms than a ring.

In particular all elements in a field, except the zero-element ($0$), must have a multiplicative inverse.

And multiplication in a field must be commutative.

Depending on the text you have, a ring may need to have a multiplicative identity ($1$) or not. Either way, a field must have it, and it must be distinct from the zero-element ($0$).

So for instance $\mathbb Z$ is a ring but not a field, since for instance $2$ does not have a multiplicative inverse.

And $\mathbb Q$ is a field, since every non-zero element in it does have a multiplicative inverse.

In this example of a ring ($\mathbb Z$), multiplication happens to be commutative, and $1$ exists as part of the ring. That's why this example is called a

- #3

roni1

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Thanks...

I have a question:

Why the multiplicative indentity (1) distincit from the zero-element (0)?

I have a question:

Why the multiplicative indentity (1) distincit from the zero-element (0)?

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I like Serena

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roni said:I have a question:

Why the multiplicative indentity (1) distincit from the zero-element (0)?

Formally it is because it says so in the definition of a Field:

Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.

The rationale is that a field is supposed to be a group with respect to multiplication - excluding the zero-element.

It can't be if 1 and 0 are the same element. After all, we already know that 0 does not have an inverse, which follows from the distributive properties.

The definition of a Ring does not have this restriction - assuming it does have a multiplicative identity. But then a ring is not a group with respect to multiplication to begin with.

- #5

HOI

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By the definition of 0, we have that x+ 0= x for all x. Since this a field we have the distributive law: y(x+ z)= yx+ yz. Taking z= 0, y(x+ 0)= yx+ y0. But x+ 0 = x so we have yx= yx+ y0. Since "cancelation" works in a field, y0= 0 for all y. But if 0= 1, the multiplicative identity, then y1= y. Together, those give y= 0 for all y. That is, if we allow the additive identity and multiplicative identities in a field to be the same element of the field, the field would have only a single member.

Some texts replace the condition that "\(\displaystyle 1\ne 0\)" in the definition of a field with "the field contains more that one element".

A ring is a circular or square area used in sports such as boxing, wrestling, and martial arts for matches or competitions. A field, on the other hand, is a large open space used in sports such as football, soccer, and baseball for matches or games.

Sports such as boxing, wrestling, martial arts, and some combat sports like MMA (Mixed Martial Arts) typically use a ring as their venue.

Sports such as football, soccer, baseball, lacrosse, and field hockey typically use a field as their venue.

A ring provides a smaller and more intimate setting for the audience, making it easier to follow the action. It also allows for a better view of the athletes for the audience. Additionally, the enclosed space of a ring can create a more intense and exciting atmosphere for the athletes and the audience.

A field allows for a larger playing area, which is necessary for sports that require a lot of running and movement. It also allows for a larger audience capacity, making it more suitable for popular sports events. Additionally, a field can be used for multiple sports, providing versatility and flexibility for different competitions and events.

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