Polynomial Rings (Units and Zero divisors)

In summary, the units in a ring of integers, Z4, are 1 and 3, while the zero-divisor is 2. In a ring of polynomials, Z4[x], the units are the constant polynomials 'a' where 'a' is a unit of Z4, such as 1 and 3. However, linear polynomials, such as x+1, x+3, and 3x+1, are not considered units. The same applies for zero-divisors, where linear polynomials with a constant value of 2, such as x+2 and 3x+2, are not considered zero-divisors. The constant is required when determining units and zero-divisors in
  • #1
facepalmer
7
0
Hi all,

I would just like to get some clarity on units and zero-divisors in rings of polynomials.
If I take a ring of Integers, Z4, (integers modulo 4) then I believe the units
are 1 & 3. And the zero-divisor is 2.

Units
1*1 = 1
3*3 = 9 = 1

Zero divisor
2*2 = 4 = 0

Now, If I take a ring of polynomials Z4[x], the polynomials with coefficients in Z4 and wish to find the units I believe that the units in Z4[x] are the constant polynomials 'a' where 'a' in a unit of Z4.
So, 1 and 3.

Now, are the polynomials of degree 1 in Z4[x] with constant values 1 and 3 considered units?
x+1, x+3, 3x+1?
Or are the linear polynomials never considered units? units can only be the constant polynomials?

Does the same apply for the zero-divisors in Z4[x]?
i.e. are the linear polynomials in Z4[x] with constant value 2; x+2, 3x+2, the zero-divisors?

hopefully I am making some sense to this question...

Thanks
 
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  • #2
facepalmer said:
Now, are the polynomials of degree 1 in Z4[x] with constant values 1 and 3 considered units?
x+1, x+3, 3x+1?
Or are the linear polynomials never considered units? units can only be the constant polynomials?

These three are not units but 2x + 1 is. So is 2x + 3.

Does the same apply for the zero-divisors in Z4[x]?
i.e. are the linear polynomials in Z4[x] with constant value 2; x+2, 3x+2, the zero-divisors?

Thanks
no but 2x + 2 is a zero divisor.
 
  • #3
Thanks, so the constant is required but along with the coefficient of the linear polynomial when determining units and zero-divisors then.
 
  • #4
facepalmer said:
Thanks, so the constant is required but along with the coefficient of the linear polynomial when determining units and zero-divisors then.

no. 2x is also a zero divisor as is 2x^n

But you are right for units.
 
  • #5
great, thanks for the assistance
 

1. What is a polynomial ring?

A polynomial ring is a mathematical structure that consists of polynomials with coefficients from a given ring. It is denoted by R[x], where R is the underlying ring and x is an indeterminate variable.

2. What are units in a polynomial ring?

Units in a polynomial ring are polynomials that have a multiplicative inverse. In other words, a polynomial a(x) is a unit if there exists another polynomial b(x) such that a(x)b(x) = 1. These polynomials are also called invertible polynomials.

3. How do you determine if a polynomial is a unit?

A polynomial a(x) is a unit if and only if the constant term is a unit in the underlying ring, and all other coefficients are zero divisors. In other words, all non-constant terms must be divisible by a non-zero element in the underlying ring.

4. What are zero divisors in a polynomial ring?

Zero divisors in a polynomial ring are non-zero polynomials that multiply to give the zero polynomial. In other words, two polynomials a(x) and b(x) are zero divisors if their product a(x)b(x) is equal to the zero polynomial.

5. How do you determine if a polynomial is a zero divisor?

A polynomial a(x) is a zero divisor if and only if there exists a non-zero polynomial b(x) such that a(x)b(x) = 0. In other words, the polynomial a(x) is not a unit and it can be multiplied by another non-zero polynomial to give the zero polynomial.

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