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I would just like to get some clarity on units and zero-divisors 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 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 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 zero-divisors in Z

_{4}[x]?

i.e. are the linear polynomials in Z

_{4}[x] with constant value 2; x+2, 3x+2, the zero-divisors?

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

Thanks