PRIMARY DECOMPOSITION OF SUBMODULES OF A FREE MODULE OF FINITE RANK OVER A BÉZOUT DOMAIN
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Let $R$ be a commutative ring with identity. In this paper, we characterize the prime submodules of a free $R$-module $F$ of finite rank with at most $n$ generators, when $R$ is a $\text{GCD}$ domain. Also, we show that if $R$ is a B\'ezout domain, then every prime submodule with $n$ generators is the row space of a prime matrix. Finally, we study the existence of primary decomposition of a submodule of $F$ over a B\'ezout domain and characterize the minimal primary decomposition of this submodule.
Abstract
Let $R$ be a commutative ring with identity. In this paper, we characterize the prime submodules of a free $R$-module $F$ of finite rank with at most $n$ generators, when $R$ is a $\text{GCD}$ domain. Also, we show that if $R$ is a B\'ezout domain, then every prime submodule with $n$ generators is the row space of a prime matrix. Finally, we study the existence of primary decomposition of a submodule of $F$ over a B\'ezout domain and characterize the minimal primary decomposition of this submodule.
참고문헌 (14)
- M. Alkan, "On primary decompositions and radicals of submodules", Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집), 10(1), pp.71~81 (2007), 장전수학회
- R. Gilmer, "Queen's Papers in Pure and Applied Mathematics", pp.90~ (1992), Queen's University
- S. Hedayat, "Characterization of prime submodules of a finitely generated free module over a PID", Houston J. Math, 31(1), pp.75~85 (2005)
- S. Hedayat, "Primary decomposition of submodules of a finitely generated module over a PID", Houston J. Math, 32(2), pp.369~377 (2006)
- S. Hedayat, "Prime and radical submodules of free modules over a PID", Houston J. Math, 32(2), pp.355~367 (2006)
- I. Kaplansky, "Chicago" (1974), University of Chicago Press
- M. D. Larsen, "Pure and Applied Mathematics, Vol. 43" (1971), Academic Press
- SONIA L’INNOCENTE, "THE ZIEGLER SPECTRUM OF THE RING OF ENTIRE COMPLEX VALUED FUNCTIONS", The Journal of Symbolic Logic, 84(1), pp.160~177 (2019), Cambridge University Press (CUP)[DOI]
- C. -P. Lu, "Prime submodules of modules", Comment. Math. Univ. St. Paul, 33(1), pp.61~69 (1984)
- R.L. McCasland, "Prime submodules", Communications in Algebra, 20(6), pp.1803~1817 (1992), Informa UK Limited[DOI]
- F. Mirzaei, "On Prime Submodules of a Finitely Generated Free Module Over a Commutative Ring", Communications in Algebra, 44(9), pp.3966~3975 (2016), Informa UK Limited[DOI]
- Fatemeh Mirzaei, "CHARACTERIZATION OF PRIME SUBMODULES OF A FREE MODULE OF FINITE RANK OVER A VALUATION DOMAIN", 대한수학회지, 54(1), pp.59~68 (2017), 대한수학회
- Dilek Pusat-Yilmaz, "Radicals of submodules of free modules", Communications in Algebra, 27(5), pp.2253~2266 (1999), Informa UK Limited[DOI]
- Yücel Tıraş, "A Characterization of Prime Submodules", Journal of Algebra, 212(2), pp.743~752 (1999), Elsevier BV[DOI]