On Weakly Hereditary Rings

We define a particular case of hereditary rings called weak hereditary rings. A sufficient condition for a weak hereditary be hereditary is given. We investigate the transfer of this notion in homomorphic image of rings, amalgamated duplication of a ring along a proper ideal, subring retracts, direct product of rings, and in trivial ring extensions. For the pullback constructions, we give an example showing that the transfer does not hold.