Abstract Algebra
Task #2
- When defining a ring, the following properties are used and exhibited:
- Commutative Property of Addition
- Associative Property of Addition
- Additive Identity
- Additive Inverse
- Associative Property of Multiplication
- Distributive Property
In order to prove that Z7 is a ring under the operations of addition and multiplication, then each individual property that is supposed to be exhibited must be properly demonstrated.
The first property I will demonstrate is the commutative property of addition. For this to work a,b ϵ Z7. By using the definition of the + operation on Z7, it would become:
Next, I would apply the commutative property of addition and then re-apply the definition of the + operation on Z7 to but it back.
Therefore, the part in green shows that it is commutative over Z7.
The second property I will demonstrate is the associative property of addition. For this to be true a, b, c ϵ Z7. To prove this, I will start by using the definition of + operation on Z7 back to back to allow me to then use the associative property of addition.
Since I have done this, I can now reapply the definition of + operation on z7 two more times to take my new answer back into the original form.
This means that a7 + (b7+c7) then is equal to (a7+b7)+c7, which properly demonstrates the associative property of addition over Z7.
The next property I will demonstrate is the Additive Identity property. This property states that a ϵ Z7. To prove this property I will use the definition of + operation on Z7, as I have done in the previous examples, will then show the identity property.
This would also be true by applying the same two steps to the “flipped” version of this problem.
In both scenarios, regardless of the order, it shows that zero is an additive identity for Z7, thus proving this true for this instance.
The next property I will prove is the additive inverse property. In this property, a ϵZ7 by the following:
The part above, in black, demonstrates the definition of _+ operation on Z7. The part in red shows the evaluation of the parenthesis in a resulting value of 0, which is caused by the definition of integer addition and integer subtraction. Regardless of the order in which they are written, the result is the same. Either way, it shows that every aϵZ7 has an additive inverse of (7-a)7 which demonstrates the additive inverse property as it relates to Z7.
The next property I will demonstrate is the Associative Property of Multiplication. The steps to proving this are very similar to those I used in the associative property of addition. I would begin by using the definition of x operation on Z7 two times so that I could then use the associative property of multiplication to change the groupings. Once the groupings are changed, I would then re-apply the two initial steps using the definition of x operation on Z7 to prove that the property is true. It would look as follows:
Therefore the a7 times the product of b7 and c7 is equal to the product of a7 and b7 times c7. This demonstrates that the multiplication property is associative over Z7, thus proving this property.
The final property to prove is the distributive property. To prove this property, it takes several rotating steps. The first steps is to apply the definition of + operation on Z7 and the immediately follow that by applying the definition of x operation Z7. This will allow me to then use the distributive property (of integers).
From here, I then reapply the definition of + operation on Z7 and then immediately follow it by a reapplication of the definition of x operation on Z7.
This process would be identical if the groupings were around alternative variables (such as a7 and b7) using the same steps. Therefore a7 times the sum of b7 and c7 is equal to the product of a7 and b7 plus the product of a7 and c7. This demonstrates the distributive property and that it applies to Z7.
Since Z7 does meet all of the properties as I have outlined, it means that Z7 meets all of the criteria of a ring.
- An integral domain can be defined as a commutative ring with unity. It also has to have a multiplication identity and a denominator/divisor of zero. In Part A, I showed that Z7 is a ring, since it meets all of the criteria for a ring.
When a, b ϵ Z7, the Commutative Property of multiplication can be demonstrated. Specifically, I can use the definition of x operation on Z7, then apply the Commutative property of multiplication (for integers), and then reapply the definition of x operation on Z7.
This demonstrates that commutative property of multiplication is true for Z7. This means that the ring Z7, is itself, a commutative ring.
The next test is to determine if Z1 is a ring with unity. To do this, I need to determine if it has a multiplication identity. To do that, I would use the definition of x operation on Z7 and then the identity property to prove this is true for both ways in which it can be multiplied.
In both scenarios, it can be seen that it does have a multiplication identity. This means that the Ring Z7 is a ring with unity. Based on the demonstration in the previous step, it means that Z7 is continuous and has unity.
Lastly, ring Z7 must have a denominator/divisor that is not qual. To determine this, a constant value k will be used.
Since a cannot be zero and since b cannot be zero, it means that k also cannot be zero. Based on that information, it means that a or b must be a multiple of 7 since 7 is a prime number.
The elements contained within Z7 are all less than 7. So even though a or b must be a multiple of 7, yet the elements in the ring are less than 7 then it means that neither a or b can be a multiple of 7.
Based on this, it means that (a * b)7 = 0, which means that a is zero or b is zero or both are zero. Since there are no values for a or b that would make (a * b)7 = 0, then it means that ring Z7 has no 0 in the denominator/divisor.
Since it meets all three requirements outline above, it means that Z7 is a commutative ring, a ring with unity, and a ring with no zero in the denominator/divisor. Since all three are met, it means that Z7 is an integral domain.