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cleaned up the grammar, and made the main question a little more clear.
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Neil Hoffman
Neil Hoffman
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Rings with a property similar to integral domaindomains

For an integral domain $R$ we know that, the intersection of two no zeronon-zero ideals is also non zero-zero, because the product of any two non zero elemetes-zero elements is no zeronon-zero. I want to know

Is the converse true, is there any equivalence condition over $R$ weaker thani.e. if $R$ is an integral domain under whichhas the property that the intersection of any two no zeronon-zero ideals is also non zero.-zero, is $R$ an integral domain? If not, what is a natural equivalent condition?

Rings with a property similar to integral domain

For an integral domain $R$ we know that the intersection of two no zero ideals is also non zero, because the product of two non zero elemetes is no zero. I want to know, is there any equivalence condition over $R$ weaker than $R$ is an integral domain under which the intersection of any two no zero ideals is also non zero.

Rings with a property similar to integral domains

For an integral domain $R$, the intersection of two non-zero ideals is also non-zero, because the product of any two non-zero elements is non-zero.

Is the converse true, i.e. if $R$ has the property that the intersection of any two non-zero ideals is also non-zero, is $R$ an integral domain? If not, what is a natural equivalent condition?

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Arena Robert
Arena Robert
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Rings with a property similar to integral domain

For an integral domain $R$ we know that the intersection of two no zero ideals is also non zero, because the product of two non zero elemetes is no zero. I want to know, is there any equivalence condition over $R$ weaker than $R$ is an integral domain under which the intersection of any two no zero ideals is also non zero.