ON ALMOST 2-ABSORBING PRIMARY IDEALS IN COMMUTATIVE RINGS

Various generalizations of primary ideals of a commutative ring R with 1 not equal 0 have been studied. For example, a proper ideal I of R is a 2-absorbing primary (resp., weakly 2-absorbing primary) if a, b, c is an element of R with abc is an element of I (resp., 0 not equal abc is an element of I) implies ab is an element of I or ac is an element of root I, or bc is an element of root I. In this paper, we introduce the concept of almost 2-absorbing primary ideal as a new generalization of primary ideals. A proper ideal I of R is called an almost 2-absorbing primary ideal of R whenever a, b, c is an element of R and abc is an element of I-I-2, then ab is an element of I or ac is an element of Rad(I) or bc is an element of Rad(I). A number of results concerning almost 2-absorbing primary ideals such as their properties and their relation among other primary ideals will be given. Also we will classify all rings for which every proper ideal is an almost 2-absorbing primary ideal.