arXiv subjects
Publications and source records attributed to Mark W. Rogers.
A positive integer k is a length of a polynomial if that polynomial factors into a product of k irreducible polynomials. We find the set of lengths of polynomials of the form x^n in R[x], where (R, m) is an Artinian local ring with m^2 = 0.
For a Noetherian local ring (R, m) having a finite residue field of cardinality q, we study the connections between the ideal Z(R) of R[x], which is the set of polynomials that vanish on R, and the ideal Z(m), the polynomials that vanish on m, using what we call pi-polynomials: polynomials of the form p(x) = \prod_{i = 1}^{q} (x - c_i), where c1, ..., cq is a set of representatives of the residue classes of m. When R is Henselian we prove that p(R) = m and show that a generating set for Z(R) may be obtained from a generating set for Z(m) by composing with p(x). When m is principal and has index of nilpotency e, we prove that if e \leq q then Z(m) = (x, m)^e, and if e = q + 1 then Z(m) = (x, m)^e + (x^q - m^{q - 1} x). When R is finite, we prove that Z(R) = \cap_{i = 1}^{q} Z(c_i + m) is a minimal primary decomposition. We determine when Z(R) is nonzero, regular, or principal, respectively, and do the same for Z(m). We prove that when R is complete, repeated application of p(x) + x to elements of R will produce a sequence converging to the roots of p(x). We show that Z(R) is the intersection of the principal ideals generated by the pi-polynomials.
Given a Noetherian local ring (R,m) it is shown that there exists an integer l such that R is Gorenstein if and only if some system of parameters contained in m^l generates an irreducible ideal. We obtain as a corollary that R is Gorenstein if and only if every power of the maximal ideal contains an irreducible parameter ideal.
Results are presented concerning the following question: If M is a finitely-generated module with finite local cohomologies over a Noetherian local ring (A, m), does there exist an integer n such that every parameter ideal for M contained in m^n has the same index of reducibility? We show that the answer is yes if M has dimension 1 or if M has dimension 2 and positive depth. This research is closely related to work of Goto-Suzuki and Goto-Sakurai; Goto-Sakurai have supplied an answer of yes in case M is Buchsbaum.