• Kyungpook Mathematical Journal
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• 제54권4호
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• pp.629-638
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• 2014

The concept of prime and weakly prime ideal in semigroups has been introduced by G. Szasz [4]. In this paper, we define the involution in po-${\Gamma}$-semigroups, then we extend some results on prime, semiprime and weakly prime ideals to the involution po-${\Gamma}$-semigroup S. Also, we characterize intra-regular involution po-${\Gamma}$-semigroups. We establish that in the involution po-${\Gamma}$-semigroup S such that the involution preserves the order, an ideal of S is prime if and only if it is both weakly prime and semiprime and if S is commutative, then the prime and weakly prime ideals of S coincide. Finally, we prove that if S is a po-${\Gamma}$-semigroup with order preserving involution, then the ideals of S are prime if and only if S is intra-regular.

• Kyungpook Mathematical Journal
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• 제62권3호
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• pp.425-436
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• 2022

In this paper we introduce the notion of 2-absorbing primary ideals of a commutative semigroup. We establish the relations between 2-absorbing primary ideals and prime, maximal, semiprimary and 2-absorbing ideals. We obtain various characterization theorems for commutative semigroups in which 2-absorbing primary ideals are prime, maximal, semiprimary and 2-absorbing ideals. We also study some other important properties of 2-absorbing primary ideals of a commutative semigroup.

• Korean Journal of Mathematics
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• 제13권2호
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• pp.217-221
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• 2005

Kehayopulu and Tsingelis [2] studied prime ideals of groupoids. Also the author studied prime left (right) ideals of groupoids. In this paper, we give some results on prime bi-ideals of groupoids.

• Kyungpook Mathematical Journal
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• 제50권1호
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• pp.117-122
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• 2010

In this paper, we introduce the concepts of 1-prime ideals and 2-prime ideals in semirings. We have also introduced $m_1$-system and $m_2$-system in semiring. We have shown that if Q is an ideal in the semiring R and if M is an $m_2$-system of R such that $\overline{Q}{\bigcap}M={\emptyset}$ then there exists as 2-prime ideal P of R such that Q $\subseteq$ P with $P{\bigcap}M={\emptyset}$.

• 한국지능시스템학회논문지
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• 제17권3호
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• pp.423-429
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• 2007

We define and study the fuzzy Jacobson radical of a ${\kappa}$-semiring. Also it is shown that the Jacobson radical of the quotient semiring R/FJR(R) of a ${\kappa}$-semiring by the fuzzy Jacobson radical FJR(R) is semisimple. And the algebraic properties of the fuzzy ideals FJR(R) and FJR(S) under a homomorphism from R onto S are also discussed.

• Korean Journal of Mathematics
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• 제29권2호
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• pp.305-320
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• 2021

In this paper, the relations between L-fuzzy semi-prime (respectively, L-fuzzy prime) ideals of a poset and L-fuzzy semi-prime (respectively, L-fuzzy prime) ideals of the lattice of all ideals of a poset are established. A result analogous to Separation Theorem is obtained using L-fuzzy semi-prime ideals.

• 대한수학회보
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• 제36권4호
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• pp.737-746
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• 1999

For a locally compact Abelian group G, and a commutative Banach algebra B, let $L^1$(G, B) be the Banach algebra of all Bochner integrable functions. We show that if G is noncompact and B is a semiprime Banach algebras in which every minimal prime ideal is cnotained in a regular maximal ideal, then $L^1$(G, B) contains no nontrivial separating idal. As a consequence we deduce some automatic continuity results for $L^1$(G, B).

• 대한수학회논문집
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• 제18권4호
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• pp.629-633
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• 2003

We investigate in this paper rings containing a finitely generated p-injective maximal left ideal. We show that if R is a semiprime ring containing a finitely generated p-injective maximal left ideal, then R is a left p-injective ring. Using this result we are able to give a new characterization of von Neumann regular rings with nonzero socle.

• 대한수학회지
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• 제46권5호
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• pp.997-1005
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• 2009

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and n a fixed positive integer. (i) If (d(x)y+xd(y)+d(y)x+$yd(x))^n$ = xy + yx for all x, y $\in$ I, then R is commutative. (ii) If char R $\neq$ = 2 and (d(x)y + xd(y) + d(y)x + $yd(x))^n$ - (xy + yx) is central for all x, y $\in$ I, then R is commutative. We also examine the case where R is a semiprime ring.

• 대한수학회보
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• 제58권3호
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• pp.659-668
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• 2021

Let R be a semiprime ring with maximal right ring of quotients Q_mr(R), and let n₁, n₂, …, n_k be k fixed positive integers. Suppose that R is (n₁+n₂+⋯+n_k)!-torsion free, and that f : 𝜌 → Q_mr(R) is an additive map, where 𝜌 is a nonzero right ideal of R. It is proved that if [[…[f(x), x^n₁], …], x^n_k] = 0 for all x ∈ 𝜌, then [f(x), x] = 0 for all x ∈ 𝜌. This gives the result of Beidar et al. [2] for semiprime rings. Moreover, it is also proved that if R is p-torsion, where p is a prime integer with p = Σ^k_i=1 n_i and if f : R → Q_mr(R) is an additive map satisfying [[…[f(x), x^n₁], …], x^n_k] = 0 for all x ∈ R, then [f(x), x] = 0 for all x ∈ R.