Hirano [On annihilator ideals of a polynomial ring over a noncommutative ring, J. Pure Appl. Algebra, 168 (2002), 45–52] studied relations between the set of annihilators in a ring R and the set of annihilators in a polynomial extension R[x] and introduced quasi-Armendariz rings. In this paper, we give a sufficient condition for a ring R and a monoid M such that the monoid ring R[M] is quasi-Armendariz. As a consequence we show that if R is a right APP-ring, then R[x]=(xn) and hence the trivial extension T(R,R) are quasi-Armendariz. They allow the construction of rings with a non-zero nilpotent ideal of arbitrary index of nilpotency which are quasi-Armendariz.
Habibi, M. and Moussavi, A., Annihilator properties of skew monoid rings, Comm. Algebra, 42(2) (2014), 842–852.
Moussavi A., 'Annihilator properties of skew monoid rings' (2014) 42Comm. Algebra: 842-852.
Moussavi A.Annihilator properties of skew monoid ringsComm. Algebra201442842852)| false