2021/02/09 by Nikmehr, M. J., Nikandish, R., Yassine, A.
#Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2102.06077
Let R be a commutative ring with identity. In this paper, we introduce the concept of weakly 1-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal I of R is called weakly 1-absorbing prime if for all nonunit elements a,b,c ∈ R such that 0≠ abc ∈ I, then either ab ∈ I or c ∈ I. A number of results concerning weakly 1-absorbing prime ideals and examples of weakly 1-absorbing prime ideals are given. It is proved that if I is a weakly 1-absorbing prime ideal of a ring R and 0 ≠ I1I2I3 ⊆ I for some ideals I1, I2, I3 of R such that I is free triple-zero with respect to I1I2I3, then I1I2 ⊆ I or I3⊆ I. Among other things, it is shown that if I is a weakly 1-absorbing prime ideal of R that is not 1-absorbing prime, then I3 = 0. Moreover, weakly 1-absorbing prime ideals of PID's and Dedekind domains are characterized. Finally, we investigate commutative rings with the property that all proper ideals are weakly 1-absorbing primes.