A remark on the unique factorization theorem

  title={A remark on the unique factorization theorem},
  author={M. Nagata},
  journal={Journal of The Mathematical Society of Japan},
  • M. Nagata
  • Published 1957
  • Mathematics
  • Journal of The Mathematical Society of Japan
Factorization in general rings and strictly cyclic modules.
The problem of factorization in noncommutative rings first arose in analysis, where the solution of a linear differential equation was reduced to the factorization of a linear differential operatorExpand
The syzygy theorem and the Weak Lefschetz property
This thesis consists of two research topics in commutative algebra. In the first chapter, a comprehensive analysis is given of the Weak Lefschetz property in the case of ideals generated by powersExpand
On p-radical descent of higher exponent
In the paper [8], P. Samuel has developed the theory of ^-radical descent of exponent one by making use of logarithmic derivatives. In this article we shall give a generalization of his theory to theExpand
On unique factorization domains
where u is a unit and p1, p2,. . . ,pk are primes in R. Note that the factorization is essentially unique (by the same argument used to prove uniqueness of factorization in PIDs). Note also that if RExpand
The biregular cancellation problem
The main purpose of this paper is to study the relationship of the above two problems. We apply the results of the birational cancellation problem [17, 4 and 13] to investigate the biregularExpand
Atomic rings and the ascending chain condition for principal ideals
Let R be a commutative ring. We say that R satisfies the ascending chain condition for principal ideals , or that R has property ( M ), if each ascending sequence ( a 1 ) ⊆ ( a 2 ) ⊆ … of principalExpand
Let k be an algebraically closed field of characteristic zero and B a factorial affine k-domain equipped with a locally nilpotent derivation δ. We investigate B when there exists an element z ∈ BExpand
An affine threefold containing an A$$ \mathbb{A} $$2-cylinder is studied. The existence of A$$ \mathbb{A} $$2-cylinders is almost equivalent to the existence of mutually commuting, independentExpand
What v-coprimality can do for you
LetD be an integral domain with quotient fieldK. Two elements x, y ∈ D\{0} are said to be v-coprime if xD ∩ yD = xyD. A saturated multiplicative set S ⊆ D\{0} is a splitting set of D if every x ∈Expand
Conformally Osserman manifolds of dimension 16 and a Weyl–Schouten theorem for rank-one symmetric spaces
A Riemannian manifold is called Osserman (conformally Osserman, respectively), if the eigenvalues of the Jacobi operator of its curvature tensor (Weyl tensor, respectively) are constant on the unitExpand