Displaying similar documents to “Errata to the paper 'On the ε-theorems' (Fund. Math. 43, p. 156-165)”

z⁰-Ideals and some special commutative rings

Karim Samei (2006)

Fundamenta Mathematicae

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In a commutative ring R, an ideal I consisting entirely of zero divisors is called a torsion ideal, and an ideal is called a z⁰-ideal if I is torsion and for each a ∈ I the intersection of all minimal prime ideals containing a is contained in I. We prove that in large classes of rings, say R, the following results hold: every z-ideal is a z⁰-ideal if and only if every element of R is either a zero divisor or a unit, if and only if every maximal ideal in R (in general, every prime z-ideal)...

On commutative rings whose prime ideals are direct sums of cyclics

M. Behboodi, A. Moradzadeh-Dehkordi (2012)

Archivum Mathematicum

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In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring ( R , ℳ ) , the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R -modules; (2) ℳ = ⨁ λ ∈ Λ R w λ where Λ is an index set and R / Ann ( w λ ) is a principal ideal ring for each λ ∈ Λ ; (3) Every prime ideal of R is a direct...