On weakly semiprime ideals of commutative rings (original) (raw)

Let R be a commutative ring with identity 1 = 0 and let I be a proper ideal of R. D. D. Anderson and E. Smith called I weakly prime if a, b ∈ R and 0 = ab ∈ I implies a ∈ I or b ∈ I. In this paper, we define I to be weakly semiprime if a ∈ R and 0 = a 2 ∈ I implies a ∈ I. For example, every proper ideal of a quasilocal ring (R, M) with M 2 = 0 is weakly semiprime. We give examples of weakly semiprime ideals that are neither semiprime nor weakly prime. We show that a weakly semiprime ideal of R that is not semiprime is a nil ideal of R. We show that if I is a weakly semiprime ideal of R that is not semiprime and 2 is not a zero-divisor of of R, then I 2 = {0} (and hence i 2 = 0 for every i ∈ I). We give an example of a ring R that admits a weakly semiprime ideal I that is not semiprime where i 2 = 0 for some i ∈ I. If R = R 1 × R 2 for some rings R 1 , R 2 , then we characterize all weakly semiprime ideals of R that are not semiprime. We characterize all weakly semiprime ideals of of Z m that are not semiprime. We show that every proper ideal of R is weakly semiprime if and only if either R is von Neumann regular or R is quasilocal with maximal ideal Nil(R) such that w 2 = 0 for every w ∈ Nil(R). Keywords Primary ideal • Prime ideal • Weakly prime ideal • 2-absorbing ideal • n-absorbing ideal • Semiprime • Weakly semiprime ideal