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In this paper, we study Noetherian Boolean rings. We show that if R is a Noetherian Boolean ring, then R is finite and R (Z_2)^n for some integer n ≥ 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) ∈ Z[x] of degree n ≥ 1, there are irreducible polynomials g(x) and h(x), each of degree n, such that g(x) + h(x) = f(x).
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참고문헌 (9)
참고문헌 신청
D. Cox / 2011 / Why Eisenstein proved the Eisenstein criterion and why Sch¨onemann discovered it first / Amer. Math. Monthly 118 : 3 ~ 21
D. Dummit / 2004 / Abstract Algebra, 3rd ed / John Wiley, Hoboken, NJ
D. Hayes / 1965 / A Goldbach theorem for polynomials with integral coefficients / Amer. Math. Monthly 72 : 45 ~ 46
D.D. Anderson / 1986 / Generalizations of Boolean rings, Boolean-like rings and von Neumann regular rings, Comment / Math. Univ. St. Pauli 35 : 69 ~ 76
F. Saidak / 2006 / On Goldbach’s conjecture for integer polynomials / Amer. Math. Monthly 113 : 541 ~ 545
D. Dummit / 2004 / Abstract Algebra, 3rd ed / John Wiley, Hoboken, NJ
D. Hayes / 1965 / A Goldbach theorem for polynomials with integral coefficients / Amer. Math. Monthly 72 : 45 ~ 46
D.D. Anderson / 1986 / Generalizations of Boolean rings, Boolean-like rings and von Neumann regular rings, Comment / Math. Univ. St. Pauli 35 : 69 ~ 76
F. Saidak / 2006 / On Goldbach’s conjecture for integer polynomials / Amer. Math. Monthly 113 : 541 ~ 545
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