Some factorization properties of idealization in commutative rings with zero divisors
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We study some factorization properties of the idealization $\idealztn{R}{M}$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $\idealztn{R}{M}$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $\idealztn{R}{M}$ is a BFR. We also characterize the idealization rings which are UFRs.
Abstract
We study some factorization properties of the idealization $\idealztn{R}{M}$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $\idealztn{R}{M}$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $\idealztn{R}{M}$ is a BFR. We also characterize the idealization rings which are UFRs.
참고문헌 (10)
- A. G. A˘garg¨un, "Factorization in commutative rings with zero divisors. III", Rocky Mountain J. Math, 31(1), pp.1~21 (2001)[DOI]
- D. D. Anderson, "Factorization in commutative rings with zero divisors", Rocky Mountain J. Math, 26(2), pp.439~480 (1996)[DOI]
- D. D. Anderson, "Idealization of a module", J. Commut. Algebra, 1(1), pp.3~56 (2009)[DOI]
- M. Axtell, "U-factorizations in commutative rings with zero divisors", Comm. Algebra, 30(3), pp.1241~1255 (2002)[DOI]
- M. Axtell, "Factorizations in self-idealizations of PIRs and UFRs", Boll. Unione Mat. Ital, 10(4), pp.649~670 (2017)[DOI]
- A. Bouvier, "Anneaux pr´esimplifiables", Rev. Roumaine Math. Pures Appl, 19, pp.713~724 (1974)
- A. Bouvier, "Structure des anneaux `a factorisation unique", Publ. D´ep. Math. (Lyon), 11(3), pp.39~49 (1974)
- G. W. Chang, "Factorization in the self-idealization of a PID", Boll. Unione Mat. Ital, 663(2), pp.363~377 (2013)
- D. Frohn, "ACCP rises to the polynomial ring if the ring has only finitely many associated primes", Comm. Algebra, 32(3), pp.1213~1217 (2004)[DOI]
- J. A. Huckaba, "Commutative rings with zero divisors, Monographs and Textbooks in Pure and Applied Mathematics", pp.117~ (1988), Marcel Dekker, Inc.