An ideal I of a commutative ring R is called a weakly primary ideal of R if whenever a, b ∈ R and 0 ab ∈ I, then a ∈ I or b ∈ √ I. An ideal I of R is called weakly 1-absorbing primary if whenever nonunit elements a, b, c ∈ R and 0 abc ∈ I, then ab ∈ I or c ∈ √ I. In this paper, we characterize rings over which every ideal is weakly 1-absorbing primary (resp. weakly primary). We also prove that, over a non-local reduced ring, every weakly 1-absorbing primary ideals is weakly primary