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The problem is to prove that, in a commutative ring with identity, the set of ideals in which every element is a zero divisor has a maximal element with respect the order of inclusion, and that every maximal element is prime. But I´m thinking* that considering the set of all zero-divisors, this set as I see it´s an ideal, and thus must be the only maximal element.

An ideal is defined as a subset J of the ring R , such that for every $ x\in R$ we have $ xJ \subset J $

I think that I´m wrong, only because it´s rare.

An extra question but related, there exist rings with element x, such that x is not a zero divisor nor a unit?

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    Careful: an (two sided) ideal is *usually* defined as a subgroup of the additive group of the ring $R$, such that for any $x\in R$ and $xJ,Jx\subseteq J$. A left ideal is one for which $xJ\subseteq J$, and a right ideal is one for which $Jx\subseteq J$.2013-12-19

2 Answers 2