Accurate Elements and Super-Primitive Elements over Rings

Let R R[α] be a simple algebraic extension of commutative rings with identity, and let I be the kernel of the R-algebra map R[X] → R[α] sending X to α, where R[X] is the polynomial ring in one indeterminate over R. Then we say that α is accurate over R if I is generated by nonzero polynomials of least degree in I. The main purpose of this paper is to give several conditions for α to be accurate over R. Our results generalize some work of Nagata and Mirbagheri and Ratliff. We also discuss some related topics, including consideration of relations between accurate elements and super-primitive elements.