On weakly semiprime ideals of commutative rings

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 Iweakly 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 ≠ a2∈ I implies a∈ I. For example, every proper ideal of a quasilocal ring (R, M) with M2= 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 I2= {0} (and hence i2= 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 i2≠ 0 for some i∈ I. If R= R1× R2 for some rings R1, R2, then we characterize all weakly semiprime ideals of R that are not semiprime. We characterize all weakly semiprime ideals of of Zm 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 w2= 0 for every w∈ Nil(R). © 2016, The Managing Editors.