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Let $A$ be a commutative ring with identity and $I$ an ideal of $A$ . $A$ is said to be $I$ - $c l e a n$ if for every element $a \in A$ there is an idempotent $e = e^{2} \in A$ such that $a - e$ is a unit and $a e$ belongs to $I$ . A filter of ideals, say $ℱ$ , of $A$ is Noetherian if for each $I \in ℱ$ there is a finitely generated ideal $J \in ℱ$ such that $J \subseteq I$ . We characterize $I$ -clean rings for the ideals $0$ , $n (A)$ , $J (A)$ , and $A$ , in terms of the frame of multiplicative Noetherian filters of ideals of $A$ , as well as in terms of more classical ring properties.

Classes of commutative rings characterized by going-up and going-down behavior

David E. Dobbs, Marco Fontana (1982)

Rendiconti del Seminario Matematico della Università di Padova

Coassociated primes of modules over a commutative ring.

Siamak Yassemi (1997)

Mathematica Scandinavica

Commutative graded- $S$ -coherent rings

Anass Assarrar, Najib Mahdou, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

Recently, motivated by Anderson, Dumitrescu’s $S$ -finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of $S$ -coherent rings, which is the $S$ -version of coherent rings. Let $R = ⨁_{α \in G} R_{α}$ be a commutative ring with unity graded by an arbitrary commutative monoid $G$ , and $S$ a multiplicatively closed subset of nonzero homogeneous elements of $R$ . We define $R$ to be graded- $S$ -coherent ring if every finitely generated homogeneous ideal of $R$ is $S$ -finitely presented. The purpose of this paper is to give the graded...

Commutative radical rings. I.

Tomáš Kepka, Petr Němec (2007)

Acta Universitatis Carolinae. Mathematica et Physica

Commutative radical rings. II.

Tomáš Kepka, Petr Němec (2008)

Acta Universitatis Carolinae. Mathematica et Physica

Commutative rings in which every proper ideal is maximal

Joachim Reineke (1977)

Fundamenta Mathematicae

Commutative rings whose principal ideals are annihilators

Camillo, Victor P. (1989)

Portugaliae mathematica

Conditions under which $R (x)$ and $R 〈 x 〉$ are almost Q-rings

Hani A. Khashan, H. Al-Ezeh (2007)

Archivum Mathematicum

All rings considered in this paper are assumed to be commutative with identities. A ring $R$ is a $Q$ -ring if every ideal of $R$ is a finite product of primary ideals. An almost $Q$ -ring is a ring whose localization at every prime ideal is a $Q$ -ring. In this paper, we first prove that the statements, $R$ is an almost $Z P I$ -ring and $R [x]$ is an almost $Q$ -ring are equivalent for any ring $R$ . Then we prove that under the condition that every prime ideal of $R (x)$ is an extension of a prime ideal of $R$ , the ring $R$ is a (an almost)...

Congruences and ideals in ternary rings

Ivan Chajda, Radomír Halaš, František Machala (1997)

Czechoslovak Mathematical Journal

A ternary ring is an algebraic structure $ℛ = (R; t, 0, 1)$ of type $(3, 0, 0)$ satisfying the identities $t (0, x, y) = y = t (x, 0, y)$ and $t (1, x, 0) = x = (x, 1, 0)$ where, moreover, for any $a$ , $b$ , $c \in R$ there exists a unique $d \in R$ with $t (a, b, d) = c$ . A congruence $θ$ on $ℛ$ is called normal if $ℛ / θ$ is a ternary ring again. We describe basic properties of the lattice of all normal congruences on $ℛ$ and establish connections between ideals (introduced earlier by the third author) and congruence kernels.

Currently displaying 1 – 19 of 19

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Displaying 1 – 19 of 19

Carrés cartésiens et anneaux de pseudo-valuation

Carrés cartésiens et couples d'anneaux ayant les mêmes idéaux premiers

Castelnuovo's index of regularity and reduction numbers

Characterization of Fans and Hereditarily Pythagorean Fields.

Class Groups of Localities of Rings of Invariants of Reductive Algebraic Groups.

Classes of Commutative Clean Rings