Left App - Property of Formal Power Series Rings

A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any element a ∈ R. We consider left APP-property of the skew formal power series ring R[[x;α]] where α is a ring automorphism of R. It is shown that if R is a ring satisfying descending chain condition on right annihilators then R[[x;α]] is left APP if and only if for any sequence (b0, b1, . . . ) of elements of R the ideal lR (∑∞ j=0 ∑∞ k=0 Rα (bj) ) is right s-unital. As an application we give a sufficient condition under which the ring R[[x]] over a left APP-ring R is left APP. Throughout this paper, R denotes a ring with unity. Recall that R is left principally quasi-Baer if the left annihilator of every principal left ideal of R is generated by an idempotent. Similarly, right principally quasi-Baer rings can be defined. A ring is called principally quasi-Baer if it is both right and left principally quasi-Baer. Observe that biregular rings and quasi-Baer rings (i.e. the rings over which the left annihilator of every left ideal of R is generated by an idempotent of R) are principally quasi-Baer. For more details and examples of left principally quasi-Baer rings, see [3], [1], [2], [4], and [7]. A ring R is called a right (resp. left) PP-ring if the right (resp. left) annihilator of every element of R is generated by an idempotent. R is called a PP-ring if it is both right and left PP. As a generalization of left principally quasi-Baer rings and right PP-rings, the concept of left APP-rings was introduced in [9]. A ring R is called a left APP-ring if the left annihilator lR(Ra) is right s-unital as an ideal of R for any element a ∈ R. For more details and examples of left APP-rings, see [9] and [6]. There are a lot of results concerning left principal quasi-Baerness and right PP-property of polynomial extensions of a ring. It was proved in ([2], Theorem 2.1) that a ring R is left principally quasi-Baer if and only if R[x] is left principally quasi-Baer. If all right semicentral idempotents of R are central, then it was shown in [7] that the ring R[[x]] is left principally quasi-Baer if and only if R is left principally quasi-Baer and every countable family of idempotents in R has a generalized join in I(R), the set of all idempotents of R. It was shown in [5] that R is a reduced PP-ring if and only if R[[x]] is a reduced PP-ring. In [8] the PP-property of the rings of generalized power series over a ring R has been 2000 Mathematics Subject Classification: primary 16W60; secondary 16P60.