0
$\begingroup$

I'm starting to study about the Ternary A-semirings and studying the paper of Daddi and Pawar entitle: Ideal Theory in Commutative Ternary A-semirings, which was published in Int. Math. Forum vol. 7 (2012) no. 42, pp. 2085-2091. In Theorem 3.3, they show that if $P$ is a maximal ideal with respect to $T$, then $P$ is prime. I understand that they will be prove by contrapositive proof of the definition of prime, which is likely to assume that $A\nsubseteq P, B\nsubseteq P$ and $C\nsubseteq P$. I do not understand why you assume that $P\subset A, P\subset B$ and $P\subset C$.

Please explain the reason for me to understand. Thank you very much.

  • 0
    @WimC: Thanks for the correction.2012-12-02

1 Answers 1

1

They are using a stronger (unstated, as far as I can see) result: $P$ is not prime if, and only if, there exists three ideals $A$, $B$ and $C$ of which $P$ is a strict subset, such that $[ABC]\subseteq P$.

So to prove that $P$ is prime, they show that for all ideals $A$, $B$ and $C$ that strictly contain $P$, $[ABC]\not\subseteq P$. I haven't proven that the stronger result holds, but it's true in usual (not ternary) rings, and it probably holds as well in the case of ternary rings.

  • 0
    Thank you for your answer.2012-12-02