Some Results on Factorization in Integral Domains

Abstract

The M-graded domains , which are almost Schreier are classified under the assumption that the integral closure of R is a root extension of R, where M is a torsion-free, commutative, cancellative monoid. In the case that D[M] is a commutative monoid domain it is shown that if M conical and is a root extension, then D[M] is almost Schreier if and only if M and D are almost Schreier. If R=ℤ[nω] is an order in a quadratic extension field of ℚ, it is shown that the conditions; R[X] is IDPF; R[X] is inside factorial; R[X] is almost Schreier; is a root extension; and every prime divisor of n also divides the discriminant of the extension K/ℚ; are equivalent conditions.

Key Words:

• Almost Schreier
• Atomic domain
• IDPF domain
• Inside factorial
• Pre-Schreier domain
• Root extension

2010 Mathematics Subject Classification:

• 13A02
• 13F15
• 13B22
• 11R04
• 11R11

ACKNOWLEDGMENT

The authors thank the referee for his or her careful reading and constructive comments on this paper.

Notes

Communicated by E. Puczylowski.