About weak $\pi$-rings
DOI:
https://doi.org/10.5269/bspm.51014Abstract
As in \cite{J}, a ring is called a weak $\pi$-ring if every regular principal ideal is a finite product of prime ideals.
In this paper, we establish some characterizations for weak $\pi$-rings. Also, we translate the properties weak $\pi$-ring and $(*)$-ring of $A\propto E$ in terms of a commutative ring $A$ and an $A$-module $E$.
References
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2. D. D. Anderson and L. A. Mahaney, (1988). On primary factorizations. J. Pure Appl. Algebra 54:141-154. https://doi.org/10.1016/0022-4049(88)90026-6
3. D. D. Anderson and M. Winders, Idealization of a module. J. Comm. Algebra 1 (2009), 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
4. C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extension definided by Prˆufer conditions, J. Pure App. Algebra 214 (2010), 53-60. https://doi.org/10.1016/j.jpaa.2009.04.011
5. R. W. Gilmer, Multiplicative Ideal Theory. New York: Marcel Dekker, Inc., (1972).
6. R. W. Gilmer, Noether's work in commutative ring theory, in "Emmy Noether" (J. Brewer and M. Smith, Eds.), Dekker, New York, in press.
7. S. Glaz, Commutative Coherent Rings, Springer-Verlag, Lecture Notes in Mathematics 1371 (1989). https://doi.org/10.1007/BFb0084570
8. J. A. Huckaba, Commutative Rings with Zero Divisors. New York: Marcel Dekker, Inc., (1988).
9. C. Jayaram, Almost Q-rings. Arch. Math. (Brno)(2004), 40:249-257.
10. C. Jayaram, Some characterizations of Dedekind rings. Commun. Algebra (2012), 40:206-212. https://doi.org/10.1080/00927872.2010.527881
11. C. Jayaram, Weak -rings. Comm. Algebra, 45:6, (2017) 2394-2400. https://doi.org/10.1080/00927872.2016.1233204
12. S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra 32 (2004), no. 10, 3937-3953. https://doi.org/10.1081/AGB-200027791
13. P. J. McCarthy, Principal elements of lattices of ideals. Proc. Amer. Math. Soc., (1971) 30:43-45. https://doi.org/10.1090/S0002-9939-1971-0279080-4
2. D. D. Anderson and L. A. Mahaney, (1988). On primary factorizations. J. Pure Appl. Algebra 54:141-154. https://doi.org/10.1016/0022-4049(88)90026-6
3. D. D. Anderson and M. Winders, Idealization of a module. J. Comm. Algebra 1 (2009), 3-56. https://doi.org/10.1216/JCA-2009-1-1-3
4. C. Bakkari, S. Kabbaj and N. Mahdou, Trivial extension definided by Prˆufer conditions, J. Pure App. Algebra 214 (2010), 53-60. https://doi.org/10.1016/j.jpaa.2009.04.011
5. R. W. Gilmer, Multiplicative Ideal Theory. New York: Marcel Dekker, Inc., (1972).
6. R. W. Gilmer, Noether's work in commutative ring theory, in "Emmy Noether" (J. Brewer and M. Smith, Eds.), Dekker, New York, in press.
7. S. Glaz, Commutative Coherent Rings, Springer-Verlag, Lecture Notes in Mathematics 1371 (1989). https://doi.org/10.1007/BFb0084570
8. J. A. Huckaba, Commutative Rings with Zero Divisors. New York: Marcel Dekker, Inc., (1988).
9. C. Jayaram, Almost Q-rings. Arch. Math. (Brno)(2004), 40:249-257.
10. C. Jayaram, Some characterizations of Dedekind rings. Commun. Algebra (2012), 40:206-212. https://doi.org/10.1080/00927872.2010.527881
11. C. Jayaram, Weak -rings. Comm. Algebra, 45:6, (2017) 2394-2400. https://doi.org/10.1080/00927872.2016.1233204
12. S. Kabbaj and N. Mahdou, Trivial extensions defined by coherent-like conditions, Comm. Algebra 32 (2004), no. 10, 3937-3953. https://doi.org/10.1081/AGB-200027791
13. P. J. McCarthy, Principal elements of lattices of ideals. Proc. Amer. Math. Soc., (1971) 30:43-45. https://doi.org/10.1090/S0002-9939-1971-0279080-4
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2022-12-23
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