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May 19th, 2018, 07:51 AM   #1
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field/ring

What the differences between ring and field?
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May 19th, 2018, 08:40 AM   #2
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Did you know the definitions of "field" and "ring"? In a field, every member, except the additive identity, has a multiplicative inverse. In a ring, it may happen that no member has a multiplicative identity.
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May 19th, 2018, 08:54 AM   #3
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Originally Posted by Country Boy View Post
In a ring, it may happen that no member has a multiplicative identity.

You mean there might exist a nonzero element with no multiplicative inverse.

In $\mathbb Z / 6 \mathbb Z$, the elements $1$ and $5$ are invertible yet $\mathbb Z / 6 \mathbb Z$ is not a field because for example $2$ is not invertible.

In fact if the ring has $1$, then $1$ is invertible.

Last edited by Maschke; May 19th, 2018 at 08:57 AM.
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May 19th, 2018, 08:56 PM   #4
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To elaborate slightly, rings and fields aren't mutually exclusive. Any field is also a ring. In particular, a field is a ring in which all elements are invertible.
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May 20th, 2018, 10:53 AM   #5
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To elaborate slightly, rings and fields aren't mutually exclusive. Any field is also a ring. In particular, a field is a ring in which all elements are invertible.
And is commutative.
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May 20th, 2018, 05:14 PM   #6
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I'm sure OP is thoroughly confused by now.
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May 21st, 2018, 05:20 AM   #7
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It's simply a matter of definition.

https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Field_...sic_definition

A Ring is a set of elements closed under two operations, addition and multiplication. which satisfy the following axioms.

1) Associativity of addition and multiplication: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.
2) Commutativity of addition and multiplication: a + b = b + a and a · b = b · a.
3) Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.
4) Additive inverses: for every a in F, there exists an element in F, denoted −a, called additive inverse of a, such that a + (−a) = 0.
5) Distributivity of multiplication over addition: a · (b + c) = (a · b) + (a · c)

Field: Ring plus 6)

6) Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by $\displaystyle a^{-1}$, $\displaystyle 1/a$, or $\displaystyle \frac{1}{a}$, called the multiplicative inverse of a, such that a · $\displaystyle a^{-1}$ = 1

The integers are a ring. The rational numbers are a field.
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May 21st, 2018, 01:12 PM   #8
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Quote:
Originally Posted by zylo View Post
It's simply a matter of definition.

https://en.wikipedia.org/wiki/Ring_(mathematics)
https://en.wikipedia.org/wiki/Field_...sic_definition

A Ring is a set of elements closed under two operations, addition and multiplication. which satisfy the following axioms.

1) Associativity of addition and multiplication: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.
2) Commutativity of addition and multiplication: a + b = b + a and a · b = b · a.
3) Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.
4) Additive inverses: for every a in F, there exists an element in F, denoted −a, called additive inverse of a, such that a + (−a) = 0.
5) Distributivity of multiplication over addition: a · (b + c) = (a · b) + (a · c)

Field: Ring plus 6)

6) Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by $\displaystyle a^{-1}$, $\displaystyle 1/a$, or $\displaystyle \frac{1}{a}$, called the multiplicative inverse of a, such that a · $\displaystyle a^{-1}$ = 1

The integers are a ring. The rational numbers are a field.
This is not quite right.

First, the operations are called addition and multiplication, and the notations for addition and multiplication used in algebra are used, but the operations need not be the same as those with the same name in arithmetic. They are analogs to the traditional operations in arithmetic because they follow common rules.

Second, a ring need not be commutative with respect to the "multiplication operation.

Third, a field is a commutative ring with a "multiplicative inverse" for every element except the "additive identity element."
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May 21st, 2018, 02:02 PM   #9
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Since there is alleged to be some difference of UK and US practice, I have refrained from posting before in this thread.
However the time has come to refer to the mathematical dictionary.
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File Type: jpg ring1.jpg (36.0 KB, 8 views)
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May 21st, 2018, 02:23 PM   #10
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Quote:
Originally Posted by JeffM1 View Post
Third, a field is a commutative ring with a "multiplicative inverse" for every element except the "additive identity element."
This would suggest the zero ring (the ring with just one element) is a field, which we usually don't want. Some equivalent ways to get around this: 1) define a field to be a commutative ring such that the non-zero elements form a group under multiplication; 2) repeat what you said, but add "non-zero" after "commutative".
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