On Annihilations of Ideals in Skew Monoid Rings

Abstract

Abstract. According to Jacobson [31], a right ideal is bounded if it contains a non-zero ideal, and Faith [15] called a ring strongly right bounded if every non-zero right ideal is bounded. From [30], a ring is strongly right AB if every non-zero right annihilator is bounded. In this paper, we introduce and investigate a particular class of McCoy rings which satisfy Property (A) and the conditions asked by Nielsen [42]. It is shown that for a u.p.-monoid M and σ : M → End(R) a compatible monoid homomorphism, if R is reversible, then the skew monoid ring R * M is strongly right AB. If R is a strongly right AB ring, M is a u.p.-monoid and σ : M → End(R) is a weakly rigid monoid homomorphism, then the skew monoid ring R * M has right Property (A).

$Σ$-semicommutative rings and their skew PBW extensions

“…Mohammadi et al in [41] defined a new class of rings called nil-reversible rings. A ring R is called nil-reversible if for every a ∈ R, b ∈ N (R), ab = 0 if and only if ba = 0.…”