Polynomial Rings (Units and Zero divisors)

[facepalmer]

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

 

[lavinia]

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.

 

[facepalmer]

Thanks, so the constant is required but along with the coefficient of the linear polynomial when determining units and zero-divisors then.

 

[lavinia]

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.

 

[facepalmer]

great, thanks for the assistance