# Polynomial Rings (Units and Zero divisors)

by facepalmer
Tags: divisors, polynomial, rings, units
 P: 7 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
P: 1,722
 Quote by facepalmer 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.
 P: 7 Thanks, so the constant is required but along with the coefficient of the linear polynomial when determining units and zero-divisors then.