Displaying similar documents to “On ideal theory of hoops”

A primrose path from Krull to Zorn

Marcel Erné (1995)

Commentationes Mathematicae Universitatis Carolinae

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Given a set X of “indeterminates” and a field F , an ideal in the polynomial ring R = F [ X ] is called conservative if it contains with any polynomial all of its monomials. The map S R S yields an isomorphism between the power set P ( X ) and the complete lattice of all conservative prime ideals of R . Moreover, the members of any system S P ( X ) of finite character are in one-to-one correspondence with the conservative prime ideals contained in P S = { R S : S S } , and the maximal members of S correspond to the maximal ideals contained...

Pasting topological spaces at one point

Ali Rezaei Aliabad (2006)

Czechoslovak Mathematical Journal

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Let { X α } α Λ be a family of topological spaces and x α X α , for every α Λ . Suppose X is the quotient space of the disjoint union of X α ’s by identifying x α ’s as one point σ . We try to characterize ideals of C ( X ) according to the same ideals of C ( X α ) ’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let m be an infinite cardinal. (1) Is there any ring R and I an ideal in R such that I is an irreducible intersection of m prime ideals? (2)...

Fixed-place ideals in commutative rings

Ali Rezaei Aliabad, Mehdi Badie (2013)

Commentationes Mathematicae Universitatis Carolinae

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Let I be a semi-prime ideal. Then P Min ( I ) is called irredundant with respect to I if I P P Min ( I ) P . If I is the intersection of all irredundant ideals with respect to I , it is called a fixed-place ideal. If there are no irredundant ideals with respect to I , it is called an anti fixed-place ideal. We show that each semi-prime ideal has a unique representation as an intersection of a fixed-place ideal and an anti fixed-place ideal. We say the point p β X is a fixed-place point if O p ( X ) is a fixed-place ideal. In...

Intersections of essential minimal prime ideals

A. Taherifar (2014)

Commentationes Mathematicae Universitatis Carolinae

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Let 𝒵 ( ) be the set of zero divisor elements of a commutative ring R with identity and be the space of minimal prime ideals of R with Zariski topology. An ideal I of R is called strongly dense ideal or briefly s d -ideal if I 𝒵 ( ) and I is contained in no minimal prime ideal. We denote by R K ( ) , the set of all a R for which D ( a ) ¯ = V ( a ) ¯ is compact. We show that R has property ( A ) and is compact if and only if R has no s d -ideal. It is proved that R K ( ) is an essential ideal (resp., s d -ideal) if and only if is an almost...