Fuzzy k-Primary Decomposition of Fuzzy k-Ideal in a Semiring

Abstract

In this paper, we establish that the Lasker–Noether theorem for a commutative ring may be generalized for a commutative semiring. We produce an example of an ideal in a Noetherian semiring which cannot be expressed as finite intersection of primary ideals of that semiring. But we manifest that if we consider an arbitrary k-ideal of a commutative Noetherian semiring, then it can be decomposed as finite intersection of primary k-ideals. Focus mainly on the fuzzy version of the above result, we are able to prove that in a commutative Noetherian semiring, every fuzzy k-ideal can be decomposed uniquely as finite intersection of fuzzy k-primary ideals of that semiring.

Keywords

• Semiring
• k-ideal
• Fuzzy primary ideals
• Fuzzy k-primary ideal
• Fuzzy k-primary decomposition