On ideals consisting entirely of zero divisors |
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Authors: | F Azarpanah OAS Karamzadeh Rezai A Aliabad |
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Institution: | 1. Department of Mathematics , Chamran University , Ahvaz, Iran;2. Institute for Studies in Theoretical Physics and Mathematics , Tehran |
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Abstract: | An ideal Iin a commutative ring Ris called a z°-ideal if Iconsists of zero-divisors and for each a? Ithe intersection of all minimal prime ideals containing ais contained in I.We prove that in a large class of rings, containing Noetherian reduced rings, Zero-dimensional rings, polynomials over reduced rings and C(X), every ideal consisting of zero-divisors is contained in a prime z°-ideal. It is also shown that the classical ring of quotients of a reduced ring is regular if and only if every prime z°-ideal is a minimal prime ideal and the annihilator of a f.g. ideal consisting of zero-divisors is nonzero. We observe that z°-ideals behave nicely under contractions and extensions. |
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