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On annihilators of BCK-algebras

1995; Springer Nature; Volume: 45; Issue: 4 Linguagem: Inglês

10.21136/cmj.1995.128559

1572-9141

Hamza A. S. Abujabal, Mustafa A. A. Obaid, Muhammad Aslam, A. B. Thaheem,

Rings, Modules, and Algebras

Let X be a commutative BCK-algebra and A an ideal of X. To any subset B of X we associate the set (A : B) = {x is an element of x Lambda B subset of or equal to A}, where x Lambda B = {x Lambda y: y is an element of B}. We show that (A : B) is an ideal of X and define it as the generalized annihilator of B (relative to A). If A = {0}, then (A : B) coincides with the usual annihilator of B (see for instance [4]). These and some other properties of generalized annihilators are contained in Section 3 of this paper. Section 4 contains some applications of generalized annihilators in quotient BCK-algebras and in the theory of prime ideals of BCK-algebras. Using the technique of generalized annihilators, we show that the quotient BCK-algebra of an involutory BCK-algebra is again an involutory BCK-algebra. We also obtain a characterization of prime ideals: A categorical ideal A is prime if and only if (A : B) = A (see Proposition 4.9). Section 2 contains some preliminary material for the development of our results.