ON PRINCIPALLY RIGHT 2-ABSORBING PRIMARY AND WEAKLY 2-ABSORBING PRIMARY IDEALS

Let r be a noncommutative ring. the purpose of this note is to investigate the concept of p right 2-absorbing primary (resp., weakly 2-absorbing) ideals generalizing 2-absorbing (resp., weakly 2-absorbing) ideals of noncommutative rings. from [4, proposition 1.3] we have that for commutative rings the notions of primary rings (ideals) coincide with p right primary rings (ideals). hence all results about 2-absorbing and weakly 2-absorbing primary ideals will follow as special cases from results proved in this note. we have √ i = p{v c r|v n ⊆ i for some positive integer n} and as in [5] we call √ i the pseudo radical of i. also √ 0 = √ r = p{v c r|v n = 0 for some positive integer n}. a proper ideal i of r is said to be a 2-absorbing (resp., weakly 2-absorbing) primary ideal if whenever a, b, c ∈ r such that arbrc ⊆ i (resp.0 6= arbrc ⊆ i ) then either ab ∈ i or bc ∈ √ i or ac ∈ √ i. a number of results concerning p right 2-absorbing primary (resp., weakly 2-absorbing) ideals are given, generalizing the corresponding results from commutative rings to noncommutative rings.