Some examples of weakly factorial rings
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Let D be a principal ideal domain, X be an indeter-minate over D, D[X] be the polynomial ring over D, and Rn =D[X]=(Xn) for an integer n ≥ 1. Clearly, Rn is a commutative Noe-therian ring with identity, and hence each nonzero nonunit of Rncan be written as a nite product of irreducible elements. In thispaper, we show that every irreducible element of Rn is a primaryelement, and thus every nonunit element of Rn can be written as anite product of primary elements.
Abstract
Let D be a principal ideal domain, X be an indeter-minate over D, D[X] be the polynomial ring over D, and Rn =D[X]=(Xn) for an integer n ≥ 1. Clearly, Rn is a commutative Noe-therian ring with identity, and hence each nonzero nonunit of Rncan be written as a nite product of irreducible elements. In thispaper, we show that every irreducible element of Rn is a primaryelement, and thus every nonunit element of Rn can be written as anite product of primary elements.
참고문헌 (2)
- D.D. Anderson, "On primary factorizations", J. Pure Appl. Algebra, 54, pp.141~154 (1988)
- G.W. Chang, "Factorization in the self-idealization of a PID", Boll. Unione Mat. Ital. (9), IV, pp.363~377 (2013)