Which Elements are the 0 and 1 in this Commutative Ring with Unit?

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In summary, the conversation discusses a set R with four elements: x, y, a, and b. Two tables are shown, one for addition and one for multiplication of these elements. The second table shows that x multiplied by any element equals x. It is stated that a+b=y and (a)(b)=y, and it is asked to determine which elements must be the 0 and the 1 in order for the set to be a commutative ring with unit. The poster is having trouble starting on a proof for this and also how to show that there exists units. It is suggested that showing the tables would be helpful.
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fk378
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Homework Statement


Given a set R={x,y,a,b}

There are 2 tables shown: one is the addition of x,y,a,b elements with x,y,a,b elements. The 2nd table is multiplication of each element with the other elements. (The 2nd table shows that x multiplied by anything equals x)

we have a+b=y and (a)(b)=y

Decide which elements must be the 0 and the 1, then prove that this is a commutative ring with unit.

The Attempt at a Solution


I know how to show it is commutative. I'm just having trouble starting on it.

Well if the addition of a and b equals the multiplication of a and b, then a=b=y=0. Is this right? But then since x multiplied by anything equals x, then x must also be 0. But then there is no element 1. So how can I show commutativity with this??

Also don't know how to show that there exists units.
 
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Hi fk378! :smile:

It would help if you showed us the table …

either use the code tag like this…

Code:
a b c d
e f g h
i j k l
m n o p

or use LaTeX and http://www.physics.udel.edu/~dubois/lshort2e/node56.html#SECTION00850000000000000000 [Broken] :smile:
 
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1. What is a commutative ring with unit?

A commutative ring with unit is a mathematical structure consisting of a set of elements, operations of addition and multiplication, and two special elements: the identity element for addition (also known as the unit element) and the identity element for multiplication (also known as the multiplicative identity). The operations of addition and multiplication follow the commutative property, meaning that changing the order of the elements does not change the result.

2. How is a commutative ring with unit different from a regular ring?

A commutative ring with unit is a type of ring where the operations of addition and multiplication follow the commutative property. In a regular ring, the commutative property may not hold for multiplication. Additionally, a commutative ring with unit must have an identity element for both addition and multiplication, while a regular ring may only have one or neither of these elements.

3. What is the significance of the unit element in a commutative ring with unit?

The unit element is significant because it serves as the identity element for multiplication. This means that when any element in the ring is multiplied by the unit element, the result is the original element. The unit element is also important in defining other properties of a commutative ring, such as the existence of inverses.

4. Can a commutative ring with unit have more than one unit element?

No, a commutative ring with unit can only have one unit element for addition and one unit element for multiplication. This is because the unit element must be unique and cannot be duplicated within the same ring. Additionally, having multiple unit elements would violate the commutative property.

5. What are some examples of commutative rings with unit?

Some examples of commutative rings with unit include the set of integers, the set of real numbers, and the set of complex numbers. Polynomial rings and residue class rings are also examples of commutative rings with unit. Furthermore, any finite field is also a commutative ring with unit.

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