• Journal of the Korean Mathematical Society
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• v.59 no.1
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• pp.71-85
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• 2022

In [4], the authors showed that if an h-vector (h₀, h₁, …, h_e) with h₁ = 4e - 4 and h_i ≤ h₁ is a Gorenstein sequence, then h₁ = h_i for every 1 ≤ i ≤ e - 1 and e ≥ 6. In th_is paper, we show that if an h-vector (h₀, h₁, …, h_e) with h₁ = 4e - 4, h₂ = 4e - 3, and h_i ≤ h₂ is a Gorenstein sequence, then h₂ = h_i for every 2 ≤ i ≤ e - 2 and e ≥ 7. We also propose an open question that if an h-vector (h₀, h₁, …, h_e) with h₁ = 4e - 4, 4e - 3 < h₂ ≤ (h₁)₍₁₎|⁺¹₊₁, and h₂ ≤ h_i is a Gorenstein sequence, then h₂ = h_i for every 2 ≤ i ≤ e - 2 and e ≥ 6.

• Bulletin of the Korean Mathematical Society
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• v.40 no.2
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• pp.223-234
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• 2003

This note contains the following results for a ring A : (1) A is simple Artinian if and only if A is a prime right YJ-injective, right and left V-ring with a maximal right annihilator ; (2) if A is a left quasi-duo ring with Jacobson radical J such that $_{A}$A/J is p-injective, then the ring A/J is strongly regular ; (3) A is von Neumann regular with non-zero socle if and only if A is a left p.p.ring containing a finitely generated p-injective maximal left ideal satisfying the following condition : if e is an idempotent in A, then eA is a minimal right ideal if and only if Ae is a minimal left ideal ; (4) If A is left non-singular, left YJ-injective such that each maximal left ideal of A is either injective or a two-sided ideal of A, then A is either left self-injective regular or strongly regular : (5) A is left continuous regular if and only if A is right p-injective such that for every cyclic left A-module M, $_{A}$M/Z(M) is projective. ((5) remains valid if 《continuous》 is replaced by 《self-injective》 and 《cyclic》 is replaced by 《finitely generated》. Finally, we have the following two equivalent properties for A to be von Neumann regula. : (a) A is left non-singular such that every finitely generated left ideal is the left annihilator of an element of A and every principal right ideal of A is the right annihilator of an element of A ; (b) Change 《left non-singular》 into 《right non-singular》in (a).(a).