Rings with reflexive ideals
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Let R be a ring with identity. A right ideal ideal I of a ring R is called reflexive (resp. completely reflexive) if aRb ⊆ I implies thatbRa⊆I(resp. ifab⊆Iimpliesthatba⊆I)foranya,b∈R. R is called reflexive (resp. completely reflexive) if the zero ideal of R is a reflexive ideal (resp. a completely reflexive ideal). Let K(R) (called the reflexive radical of R) be the intersection of all reflexive ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an reflexive ideal of a ring are obtained; (2) reflexive (resp. completely reflexive) property is Morita invariant; (3) For any ring R, we have K(Mn(R)) = Mn(K(R)) where Mn(R) is the ring of all n by n matrices over R; (4) For a ring R, we have K(R)[x] ⊆ K(R[x]); in particular, if R is quasi-Armendaritz, then R is reflexive if and only if R[x] is reflexive.
Abstract
Let R be a ring with identity. A right ideal ideal I of a ring R is called reflexive (resp. completely reflexive) if aRb ⊆ I implies thatbRa⊆I(resp. ifab⊆Iimpliesthatba⊆I)foranya,b∈R. R is called reflexive (resp. completely reflexive) if the zero ideal of R is a reflexive ideal (resp. a completely reflexive ideal). Let K(R) (called the reflexive radical of R) be the intersection of all reflexive ideals of R. In this paper, the following are investigated: (1) Some equivalent conditions on an reflexive ideal of a ring are obtained; (2) reflexive (resp. completely reflexive) property is Morita invariant; (3) For any ring R, we have K(Mn(R)) = Mn(K(R)) where Mn(R) is the ring of all n by n matrices over R; (4) For a ring R, we have K(R)[x] ⊆ K(R[x]); in particular, if R is quasi-Armendaritz, then R is reflexive if and only if R[x] is reflexive.
참고문헌 (8)
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