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Ideal theory in the ternary semiring \(\mathbb{Z}^-_0\). (English) Zbl 1208.16039

From the introduction: T. K. Dutta and S. Kar [Advances in algebra. Proceedings of the ICM satellite conference in algebra and related topics, Hong Kong, China, 2002. River Edge: World Scientific. 343-355 (2003; Zbl 1041.16040)], have introduced the notion of ternary semiring which generalizes the notion of ternary ring introduced by W. G. Lister [Trans. Am. Math. Soc. 154, 37-55 (1971; Zbl 0216.06901)]. The set \(\mathbb{Z}^-_0\) of all non-positive integers is an example of a ternary semiring with usual binary addition and ternary multiplication.
Our main purpose of this paper is to study the ideal theory in the ternary semiring \(\mathbb{Z}^-_0\). In Section 2, we give some basic definitions and examples. In Section 3, we study the ideal theory in the ternary semiring \(\mathbb{Z}^-_0\) and prove that \(\mathbb{Z}^-_0\) is a Noetherian ternary semiring and also an almost principal ideal ternary semiring.

MSC:

16Y60 Semirings
16D25 Ideals in associative algebras
17A40 Ternary compositions
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