
#1
Feb1812, 08:43 AM

P: 7

Hi all,
I would just like to get some clarity on units and zerodivisors in rings of polynomials. If I take a ring of Integers, Z_{4}, (integers modulo 4) then I believe the units are 1 & 3. And the zerodivisor is 2. Units 1*1 = 1 3*3 = 9 = 1 Zero divisor 2*2 = 4 = 0 Now, If I take a ring of polynomials Z_{4}[x], the polynomials with coefficients in Z_{4} and wish to find the units I believe that the units in Z_{4}[x] are the constant polynomials 'a' where 'a' in a unit of Z_{4}. So, 1 and 3. Now, are the polynomials of degree 1 in Z_{4}[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 zerodivisors in Z_{4}[x]? i.e. are the linear polynomials in Z_{4}[x] with constant value 2; x+2, 3x+2, the zerodivisors? hopefully I am making some sense to this question... Thanks 



#2
Feb1812, 09:44 AM

Sci Advisor
P: 1,716

no but 2x + 2 is a zero divisor. 



#3
Feb1912, 07:55 AM

P: 7

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




#4
Feb1912, 09:37 AM

Sci Advisor
P: 1,716

Polynomial Rings (Units and Zero divisors)But you are right for units. 



#5
Feb1912, 01:56 PM

P: 7

great, thanks for the assistance



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