Primary Decomposition of Ideals in Polynomial Rings
Guang Fu, Robert Gilmer · 2023
Our work in this paper was motivated by a question raised by D. D. Anderson (see [ A1 ], [ A2 ]). Specifically, Anderson asked whether the condition that each of its finitely generated ideals admits a finite primary decomposition is inherited by the polynomial ring R [ x ] from the coefficient ring R . In exploring the question we encountered general questions concerning finite primary decomposition that seemed at least as interesting as the original question, and these answers would be helpful in attacking the original. Perhaps the most fundamental problem in this regard is that no easily applicable criteria are known for determining whether a fixed ideal of a commutative ring admits a finite primary decomposition. (Krull raised the problem of determining such criteria in [ Kr , p. 12], and the problem is addressed in [ GH2 , Section 1] and in [ F ].)