Polynomial Rings (Units and Zero divisors)

  1. 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
     
  2. jcsd
  3. lavinia

    lavinia 2,052
    Science Advisor

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


    no but 2x + 2 is a zero divisor.
     
  4. Thanks, so the constant is required but along with the coefficient of the linear polynomial when determining units and zero-divisors then.
     
  5. lavinia

    lavinia 2,052
    Science Advisor

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

    But you are right for units.
     
  6. great, thanks for the assistance
     
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