DOI QR코드

DOI QR Code

REMARKS ON A GOLDBACH PROPERTY

  • Received : 2011.10.13
  • Accepted : 2011.11.30
  • Published : 2011.12.30

Abstract

In this paper, we study Noetherian Boolean rings. We show that if R is a Noetherian Boolean ring, then R is finite and $R{\simeq}(\mathbb{Z}_2)^n$ for some integer $n{\geq}1$. If R is a Noetherian ring, then R/J is a Noetherian Boolean ring, where J is the intersection of all ideals I of R with |R/I| = 2. Thus R/J is finite, and hence the set of ideals I of R with |R/I| = 2 is finite. We also give a short proof of Hayes's result : For every polynomial $f(x){\in}\mathbb{Z}[x]$ of degree $n{\geq}1$, there are irreducible polynomials $g(x)$ and $h(x)$, each of degree $n$, such that $g(x)+h(x)=f(x)$.

References

  1. D.D. Anderson, Generalizations of Boolean rings, Boolean-like rings and von Neumann regular rings, Comment. Math. Univ. St. Pauli 35 (1986), 69-76.
  2. D. Cox, Why Eisenstein proved the Eisenstein criterion and why Schonemann discovered it first, Amer. Math. Monthly 118 (2011), 3-21. https://doi.org/10.4169/amer.math.monthly.118.01.003
  3. D. Dummit and R. Foots, Abstract Algebra, 3rd ed., John Wiley, Hoboken, NJ, 2004.
  4. R. Gilmer, Background and preliminaries on zero-dimensional rings, Lect. Notes Pure Appl. Math. 171 (1994), 1-13.
  5. R. Gilmer, Zero-dimensionality and products of commutative rings, Lect. Notes Pure Appl. Math. 171 (1994), 15-25.
  6. D. Hayes, A Goldbach theorem for polynomials with integral coefficients, Amer. Math. Monthly 72 (1965), 45-46. https://doi.org/10.2307/2312999
  7. P. Pollack, On polynomial rings with a Goldbach property, Amer. Math. Monthly 118 (2011), 71-77.
  8. F. Saidak, On Goldbach's conjecture for integer polynomials, Amer. Math. Monthly 113 (2006), 541-545. https://doi.org/10.2307/27641978
  9. P. Samuel, About Euclidean rings, J. Algebra 19 (1971), 282-301. https://doi.org/10.1016/0021-8693(71)90110-4