ON THE STRUCTURE OF ZERO-DIVISOR ELEMENTS IN A NEAR-RING OF SKEW FORMAL POWER SERIES

The main purpose of this paper is to study the zero-divisor properties of the zero-symmetric near-ring of skew formal power series RoP; all, where R is a symmetric, a-compatible and right Noetherian ring. It is shown that if R is reduced, then the set of all zero-divisor elements of Ro [[x; a]] forms an ideal of Ro [[x; a]] if and only if Z(R) is an ideal of R. Also, if R is a non-reduced ring and ann R (a b) fl Nil (R) 0 for each a, b E Z(R), then Z (Ro [[x; a]1) is an ideal of Ro [[x; a]l. Moreover, if R is a non-reduced right Noetherian ring and Z(RoP; a11) forms an ideal, then ammR(a b) fl Nil(R) 0 for each a, b E Z(R). Also, it is proved that the only possible diameters of the zero-divisor graph of Ro[[x; a]] is 2 and 3.