A remark on prime ideals

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S.C. Lee
R. Varmazyar

Abstract

If M is a torsion-free module over an integral domain, then we show that for each submodule N of M the envelope EM (N ) of N in M is an essential extension of N. In particular, if N is divisible then EM (N ) = N . The last condition says that N is a semiprime submodule of M if N is proper.
Let M be a module over a ring R such that for any ideals a, b of R, (a ∩ b)M = aM ∩ bM . If N is an irreducible and weakly semiprime submodule of M , then we prove that (N :R M ) is a prime ideal of R. As a result, we obtain that if p is an irreducible ideal of a ring R such that a2 ⊆ p (a is an ideal of R) ⇒ a ⊆ p, then p is a prime ideal.

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How to Cite
Lee, S., & Varmazyar, R. (2020). A remark on prime ideals. Extracta Mathematicae, 35(1), 43-54. Retrieved from https://publicaciones.unex.es/index.php/EM/article/view/2605-5686.35.1.43
Section
Commutative Algebra

References

[1] A. Barnard, Multiplication modules, J. Algebra 71 (1) (1981), 174 – 178.
[2] D. Eisenbud, “ Commutative Algebra ”, Graduate Texts in Mathematics 150, Springer-Verlag, New York, 1995.
[3] Z. El-Bast, P.F. Smith, Multiplication modules, Comm. Algebra 16 (4) (1988), 755 – 779.
[4] J. Jenkins, P.F. Smith, On the prime radical of a module over a commutative ring, Comm. Algebra 20 (12) (1992), 3593 – 3602.
[5] S.T. Hu, “ Introduction to Homological Algebra ”, Holden-Day, Inc., San Francisco-London-Amsterdam, 1968.
[6] T.Y. Lam, “ Lectures on Modules and Rings ”, Graduate Texts in Mathematics 189, Springer-Verlag, New York, 1999.
[7] S.C. Lee, D.S. Lee, Direct sums of indecomposable injective modules, Bull. Austral. Math. Soc. 62 (1) (2000), 57 – 66.
[8] S.C. Lee, R. Varmazyar, Semiprime submodule of a module and related concepts, J. Algebra Appl. 18 (8) (2019), 1950147, 11 pp.
[9] R.L. McCasland, P.F. Smith, On isolated submodules, Comm. Algebra 34 (8) (2006), 2977 – 2988.
[10] D.W. Sharpe, P. Vámos, “ Injective Modules ”, Cambridge Tracts in Mathematics and Mathematical Physics 62, Cambridge University Press, London-New York, 1972.
[11] H.A. Tavallaee, R. Varmazyar, Semi-radicals of submodules in modules, I. J. Engineering Science 19 (1) (2008), 21 – 27.
[12] F. Wang, H. Kim, Two generalizations of projective modules and their applications, J. Pure Appl. Algebra 219 (6) (2015), 2099 – 2123.