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Research Article

On the lattice of annihilator ideals and its applications

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Pages 2444-2456 | Received 01 Aug 2019, Accepted 04 Jan 2021, Published online: 21 Jan 2021
 

Abstract

In this paper we study the properties of rAnn(id(R)) (the set of right annihilator ideals of R) as a lattice. We prove that (rAnn(id(R)),) is a Boolean algebra if and only if R is a semiprime ring. For a quasi Armandariz ring R, we show that the lattices rAnn(id(R)) and rAnn(id(R[x])) are isomorphic. In general, for any ring R, we show that the lattices rAnn(id(R)), rAnn(id(Mn(R))) and rAnn(id(Tn(R))) are isomorphic. We answer two questions related to property (a.c.) that were raised by Hong, Kim, Lee and Nielsen. To do this, we introduce a new class of rings. We say R is right strongly quasi Armandariz (s.q.-Armandariz) if for each f(x)R[x], there exists aR such that rR[x](f(x)R[x])=rR(aR)R[x]. The class of right s.q.-Armandariz rings is between right p.q.-Baer rings and Armandariz rings. For a right s.q.-Armandariz ring R, it is proved that R has the right annihilator condition (a.c.) if and only if R[x] has right the annihilator condition (a.c.). We conclude that if R is a quasi Armandariz ring and has right (a.c.), then R[x] has right (a.c.).

2000 Mathematics Subject Classification:

Acknowledgements

The authors are grateful to the referee for suggestions that helped improve the presentation of the paper.

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