S-PRIMAL IDEALS IN TRIVIAL EXTENSION AND IN AMALGAMATION ALGEBRA
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Let A be a commutative ring with identity and S be a multiplicative subset of A. In this paper, we introduce and study the notion of S-primal ideals as a generalization of the notion of primal ideals. We define an ideal I disjoint with S to be S-primal if there exists s ∈ S such that (I : s) is primal. Several properties of S-primal ideals are given. We investigate the behavior of the S-primal ideals under passage to some algebraic constructions such as homomorphic image, direct product and localization. Also, we study the relationship between the S-primal ideals, S-irreducible and S-primary ideals. Moreover, we examine the transfer of S-primal property to some ideals of trivial ring extensions and amalgamation of rings along an ideal.
Abstract
Let A be a commutative ring with identity and S be a multiplicative subset of A. In this paper, we introduce and study the notion of S-primal ideals as a generalization of the notion of primal ideals. We define an ideal I disjoint with S to be S-primal if there exists s ∈ S such that (I : s) is primal. Several properties of S-primal ideals are given. We investigate the behavior of the S-primal ideals under passage to some algebraic constructions such as homomorphic image, direct product and localization. Also, we study the relationship between the S-primal ideals, S-irreducible and S-primary ideals. Moreover, we examine the transfer of S-primal property to some ideals of trivial ring extensions and amalgamation of rings along an ideal.
참고문헌 (19)
- D.D. Anderson, "Idealization of a module", J. Commut. Algebra, 1, pp.3~56 (2009)
- A. Anebri, "On (1, r)-ideals of commutative rings", J. Algebra Appl, 23(2), pp.2450023~ (2024)[DOI]
- A. Anebri, "On modules satisfying the descending chain condition on r-submodules", Comm. Algebra, 50(1), pp.383~391 (2022)
- S.E. Atani, "Some remarks on primal submodules", Sarajevo Journal of Mathematics
- M.F. Atiyah, "Introduction to Commutative Algebra" (1969), Addison-Wesley Publishing Company
- M. D’Anna, "An amalgamated duplication of a ring along an ideal:the basic properties", J. Algebra Appl, 6(3), pp.443~459 (2007)
- M. D’Anna, "Amalgamated algebras along an ideal", Comm. Algebra and Applications, pp.155~172 (2009)
- M. D’Anna, "Properties of chains of prime ideals in amalgamated algebras along an ideal", J. Pure Applied Algebra, 214, pp.1633~1641 (2010)
- J. Dauns, "Primal modules", Comm. Algebra, 25(8), pp.2409~2435 (1997)
- T. Dumitrescu, "Radical factorization for trivial extensions and amalgamated duplication rings", J. Algebra Appl, 20(02), pp.2150025~ (2021)
- S. Ebrahimi Atani, "On weakly primal ideals (I)", Demonstratio Math, 40(1), pp.23~32 (2007)
- L. Fuchs, "On primal ideals", Proc. Amer. Math. Soc, 1, pp.1~6 (1950)
- L. Fuchs, "Ideal theory in Prufer domains", J. Algebra, 252, pp.411~430 (2002)
- A. Hamed, "S-prime ideals of a commutative ring", Beitr Algebra Geom, pp.1~10 (2019)
- J. Huckaba, "Commutative Rings with Zero Divisors" (1988), Dekker
- I. Kaplansky, "Commutative Algebra", pp.1974
- M. Nagata, "Local Rings" (1962), Wiley-Interscience
- S. Visweswaran, "Some results on S-primary ideals of a commutative ring", Beitr Algebra Geom
- O. Zariski, "commutative Algebra, V. 1" (1960), D. Van Nostra nd Company, inc