On Ideals of Polynomial Rings
On Ideals of Polynomial Rings M.Velrajan Let R be a commutative ring with identity and R[x] be the set of all polynomials in an indeterminate x, with coefficients in R. Then R[x] is also a commutative ring with identity. Necessary and sufficient conditions for a polynomial in x over R to be a unit, a nilpotent element and to be a zero divisor of the ring of polynomials R[x] are discussed in the post : https://velrajanm.blogspot.com/2025/03/polynomial-rings-units-nilpotents-and.html . Now we discuss some relations between the ideals of R and the ideals of R[x]. W e prove that for each non negative integer m ≥ 0, the map I → I {m} [x] is a one to one map that preserves the order ⊆ from the set of all ideals of R onto the set of ideals of R[x] that contain {a 0 + a 1 x + ∙ ∙ ∙ + a m-1 x m-1 + a m+1 x m+1 + ∙ ∙ ∙ + a n x n / a i ∈ R and n ≥ 0}, the ideal of all polynomials in R[x] with coefficient of x m = 0, where I {m} [x] = {a 0 + a 1 x +...
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