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 May 19th, 2018, 08:51 AM #1 Senior Member   Joined: Nov 2011 Posts: 247 Thanks: 3 field/ring What the differences between ring and field?
 May 19th, 2018, 09:40 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 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.
May 19th, 2018, 09:54 AM   #3
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Quote:
 Originally Posted by Country Boy 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 09:57 AM.

 May 19th, 2018, 09:56 PM #4 Senior Member   Joined: Sep 2016 From: USA Posts: 531 Thanks: 304 Math Focus: Dynamical systems, analytic function theory, numerics 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.
May 20th, 2018, 11:53 AM   #5
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Quote:
 Originally Posted by SDK 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.

 May 20th, 2018, 06:14 PM #6 Senior Member   Joined: Aug 2012 Posts: 2,126 Thanks: 618 I'm sure OP is thoroughly confused by now. Thanks from studiot and Joppy
 May 21st, 2018, 06:20 AM #7 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,621 Thanks: 117 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.
May 21st, 2018, 02:12 PM   #8
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Quote:
 Originally Posted by zylo 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."

May 21st, 2018, 03:02 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.
Attached Images
 field1.jpg (36.7 KB, 8 views) ring1.jpg (36.0 KB, 8 views)

May 21st, 2018, 03:23 PM   #10
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Quote:
 Originally Posted by JeffM1 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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