May 19th, 2018, 07:51 AM  #1 
Senior Member Joined: Nov 2011 Posts: 197 Thanks: 2  field/ring
What the differences between ring and field?

May 19th, 2018, 08:40 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,191 Thanks: 871 
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, 08:54 AM  #3  
Senior Member Joined: Aug 2012 Posts: 1,919 Thanks: 533  Quote:
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.  
May 19th, 2018, 08:56 PM  #4 
Senior Member Joined: Sep 2016 From: USA Posts: 383 Thanks: 206 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, 10:53 AM  #5 
Senior Member Joined: Oct 2009 Posts: 406 Thanks: 140  
May 20th, 2018, 05:14 PM  #6 
Senior Member Joined: Aug 2012 Posts: 1,919 Thanks: 533 
I'm sure OP is thoroughly confused by now.

May 21st, 2018, 05:20 AM  #7 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,364 Thanks: 100 
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, 01:12 PM  #8  
Senior Member Joined: May 2016 From: USA Posts: 1,038 Thanks: 423  Quote:
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, 02:02 PM  #9 
Senior Member Joined: Jun 2015 From: England Posts: 829 Thanks: 244 
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. 
May 21st, 2018, 02:23 PM  #10 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 197 Thanks: 59 Math Focus: Algebraic Number Theory, Arithmetic Geometry  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 nonzero elements form a group under multiplication; 2) repeat what you said, but add "nonzero" after "commutative".


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