Nearring of ore extensions: Zero-divisors and regular elements

Abstract

Let [Formula: see text] be a symmetric and [Formula: see text]-compatible ring. In this paper, we first specify the zero-divisor elements of the near-ring [Formula: see text]. We prove that if [Formula: see text] is an [Formula: see text]-rigid ring, then the set of all zero-divisor elements of the near-ring [Formula: see text] forms an ideal of [Formula: see text] if and only if [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text] has the right Property [Formula: see text]. Also, we are interested in studying the zero-divisor graph of [Formula: see text] which is denoted by [Formula: see text]. It is shown that [Formula: see text], and if [Formula: see text] is not reduced, then [Formula: see text] for each [Formula: see text] if and only if [Formula: see text]. Moreover, we characterize the units, the clean elements, the regular elements, the nilpotent elements and also the [Formula: see text]-regular elements of the near-ring [Formula: see text], where [Formula: see text] is a reversible and [Formula: see text]-compatible ring and [Formula: see text] is a countable locally nilpotent ideal of [Formula: see text]. Finally, we prove that the set of all [Formula: see text]-regular elements of [Formula: see text] forms a semigroup, where [Formula: see text] is a reversible and [Formula: see text]-compatible ring and [Formula: see text] is a countable locally nilpotent ideal of [Formula: see text].