Polynomial Rings - Units, Nilpotents and Zero divisors
Polynomial Rings - Units, Nilpotents and Zero divisors M.Velrajan Let R be a commutative ring with identity and R[x] be the ring of polynomials in an indeterminate x, with coefficients in R. We find necessary and sufficient conditions for a polynomial in x over R to be a unit, a nilpotent element and a zero divisor of the ring of polynomials R[x]. And we prove that the nilradical of R[x] = the Jacobson radical of R[x]. Throughout R denotes a commutative ring with identity. First we prove the property of nilpotent elements of R. 1. Let a be a nilpotent element in R. Then 1 + a is a unit in R. And in general, the sum of a unit and a nilpotent element is a unit. Proof Since a is a nilpotent element in R, a n = 0 for some integer n > 0. To prove 1 + a is a unit, we have to find y such that (1 + a)y = 1 using a n = 0. Think!! What is the relation between 1 + a, a n and 1. Yes, the simple algebraic relation . . . (1 + a) (1 - a + a 2 -...