The rings which are Boolean

Ivan Chajda; Filip Švrček

Displaying similar documents to “The rings which are Boolean”

Notes on generalizations of Bézout rings

Haitham El Alaoui, Hakima Mouanis (2021)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we give new characterizations of the $P$ - $2$ -Bézout property of trivial ring extensions. Also, we investigate the transfer of this property to homomorphic images and to finite direct products. Our results generate original examples which enrich the current literature with new examples of non- $2$ -Bézout $P$ - $2$ -Bézout rings and examples of non- $P$ -Bézout $P$ - $2$ -Bézout rings.

Semicommutativity of the rings relative to prime radical

Handan Kose, Burcu Ungor (2015)

Commentationes Mathematicae Universitatis Carolinae

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In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$ -semicommutative. We prove that a ring $R$ is $P$ -semicommutative if and only if $R [x]$ is $P$ -semicommutative if and only if $R [x, x^{- 1}]$ is $P$ -semicommutative. Also, if $R [[x]]$ is $P$ -semicommutative, then $R$ is $P$ -semicommutative. The converse holds provided that $P (R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$ , $R$ is $P$ -semicommutative...

A subclass of strongly clean rings

Orhan Gurgun, Sait Halicioglu and Burcu Ungor (2015)

Communications in Mathematics

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In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$ , and let $J^{#}$ denote the set of all elements of $R$ which are nilpotent in $R / J$ . An element $a \in R$ is called provided that there exists an idempotent $e \in R$ such that $a e = e a$ and $a - e$ or $a + e$ is an element of $J^{#}$ . A ring $R$ is said to be in case every element in $R$ is very $J^{#}$ -clean. We prove that every very $J^{#}$ -clean ring is strongly $π$ -rad clean and has stable range one. It is shown that for a...

Certain decompositions of matrices over Abelian rings

Nahid Ashrafi, Marjan Sheibani, Huanyin Chen (2017)

Czechoslovak Mathematical Journal

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A ring $R$ is (weakly) nil clean provided that every element in $R$ is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let $R$ be abelian, and let $n \in ℕ$ . We prove that $M_{n} (R)$ is nil clean if and only if $R / J (R)$ is Boolean and $M_{n} (J (R))$ is nil. Furthermore, we prove that $R$ is weakly nil clean if and only if $R$ is periodic; $R / J (R)$ is $ℤ_{3}$ , $B$ or $ℤ_{3} \oplus B$ where $B$ is a Boolean ring, and that $M_{n} (R)$ is weakly nil clean if and only if $M_{n} (R)$ is nil clean for all $n \geq 2$ .

Avoidance principle and intersection property for a class of rings

Rahul Kumar, Atul Gaur (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with identity. If a ring $R$ is contained in an arbitrary union of rings, then $R$ is contained in one of them under various conditions. Similarly, if an arbitrary intersection of rings is contained in $R$ , then $R$ contains one of them under various conditions.

Strongly 2-nil-clean rings with involutions

Huanyin Chen, Marjan Sheibani Abdolyousefi (2019)

Czechoslovak Mathematical Journal

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A $*$ -ring $R$ is strongly 2-nil- $*$ -clean if every element in $R$ is the sum of two projections and a nilpotent that commute. Fundamental properties of such $*$ -rings are obtained. We prove that a $*$ -ring $R$ is strongly 2-nil- $*$ -clean if and only if for all $a \in R$ , $a^{2} \in R$ is strongly nil- $*$ -clean, if and only if for any $a \in R$ there exists a $*$ -tripotent $e \in R$ such that $a - e \in R$ is nilpotent and $e a = a e$ , if and only if $R$ is a strongly $*$ -clean SN ring, if and only if $R$ is abelian, $J (R)$ is nil and $R / J (R)$ is $*$ -tripotent. Furthermore, we explore...

Coherent ultrafilters and nonhomogeneity

Jan Starý (2015)

Commentationes Mathematicae Universitatis Carolinae

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We introduce the notion of a coherent $P$ -ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a $P$ -point on $ω$ , and show that these ultrafilters exist generically under $𝔠 = 𝔡$ . This improves the known existence result of Ketonen [On the existence of $P$ -points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1,...

On weakened $(α, δ)$ -skew Armendariz rings

Alireza Majdabadi Farahani, Mohammad Maghasedi, Farideh Heydari, Hamidagha Tavallaee (2022)

Mathematica Bohemica

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In this note, for a ring endomorphism $α$ and an $α$ -derivation $δ$ of a ring $R$ , the notion of weakened $(α, δ)$ -skew Armendariz rings is introduced as a generalization of $α$ -rigid rings and weak Armendariz rings. It is proved that $R$ is a weakened $(α, δ)$ -skew Armendariz ring if and only if $T_{n} (R)$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz if and only if $R [x] / (x^{n})$ is weakened $(\bar{α}, \bar{δ})$ -skew Armendariz ring for any positive integer $n$ .

Remarks on $L B I$ -subalgebras of $C (X)$

Mehdi Parsinia (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $A (X)$ denote a subalgebra of $C (X)$ which is closed under local bounded inversion, briefly, an $L B I$ -subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A (X)$ , we generalize the notion of $z_{A}^{β}$ -ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C (X)$ ...

On atomic ideals in some factor rings of $C (X, ℤ)$

Alireza Olfati (2021)

Commentationes Mathematicae Universitatis Carolinae

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A nonzero $R$ -module $M$ is atomic if for each two nonzero elements $a, b$ in $M$ , both cyclic submodules $R a$ and $R b$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$ -space $X$ , the factor rings $C (X, ℤ) / C_{F} (X, ℤ)$ and $C_{c} (X) / C_{F} (X)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$ , the factor ring $ℤ^{X} / ℤ^{(X)}$ has no atomic ideal. Another result is that for each infinite...