WEAKLY PRIME AND WEAKLY COMPLETELY PRIME IDEALS OF NONCOMMUTATIVE RINGS

Year 2020, Volume: 28 Issue: 28, 43 - 60, 14.07.2020

Nico Groenewald

https://doi.org/10.24330/ieja.768127

Cited By: 5

Abstract

Anderson-Smith studied weakly prime ideals for a commutative ring
with identity. Hirano, Poon and Tsutsui studied the structure of
a ring in which every ideal is weakly prime for rings, not
necessarily commutative. In this note we give some more properties
of weakly prime ideals in noncommutative rings. We introduce the
notion of a weakly prime radical of an ideal. We initiate the
study of weakly completely prime ideals and investigate
rings for which every proper ideal is weakly completely prime.

Keywords

Weakly prime ideal, weakly prime radical, weakly completely prime ideal

References

• D. D. Anderson and E. Smith, Weakly prime ideals, Houston J. Math., 29(4) (2003), 831-840.
• A. Badawi, U. Tekir and E. Yetkin, On weakly 2-absorbing primary ideals of commutative rings, J. Korean Math. Soc., 52(1) (2015), 97-111.
• P. K. Beiranvand and R. Beyranvand, Almost prime and weakly prime submodules, J. Algebra Appl., 18(7) (2019), 1950129 (14 pp).
• G. Birkenmeier, H. Heatherly and E. Lee, Prime ideals and prime radicals in near-rings, Monatsh. Math., 117 (1994), 179-197.
• Y. Hirano, E. Poon and H. Tsutsui, On rings in which every ideal is weakly prime, Bull. Korean Math. Soc., 47(5) (2010), 1077-1087.
• N. H. McCoy, A note on finite unions of ideals and subgroups, Proc. Amer. Math. Soc., 8 (1957), 633-637.
• S. Veldsman, A note on the radicals of idealizations, Southeast Asian Bull. Math., 32 (2008), 545-551.