Displaying similar documents to “The Bordalo order on a commutative ring”

A generating family for the Freudenthal compactification of a class of rimcompact spaces

Jesús M. Domínguez (2003)

Fundamenta Mathematicae

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For X a Tikhonov space, let F(X) be the algebra of all real-valued continuous functions on X that assume only finitely many values outside some compact subset. We show that F(X) generates a compactification γX of X if and only if X has a base of open sets whose boundaries have compact neighborhoods, and we note that if this happens then γX is the Freudenthal compactification of X. For X Hausdorff and locally compact, we establish an isomorphism between the lattice of all subalgebras...

Nil-clean and unit-regular elements in certain subrings of 𝕄 2 ( ℤ )

Yansheng Wu, Gaohua Tang, Guixin Deng, Yiqiang Zhou (2019)

Czechoslovak Mathematical Journal

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An element in a ring is clean (or, unit-regular) if it is the sum (or, the product) of an idempotent and a unit, and is nil-clean if it is the sum of an idempotent and a nilpotent. Firstly, we show that Jacobson’s lemma does not hold for nil-clean elements in a ring, answering a question posed by Koşan, Wang and Zhou (2016). Secondly, we present new counter-examples to Diesl’s question whether a nil-clean element is clean in a ring. Lastly, we give new examples of unit-regular elements...

The Rings Which Can Be Recovered by Means of the Difference

Ivan Chajda, Filip Švrček (2013)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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It is well known that to every Boolean ring ℛ can be assigned a Boolean algebra ℬ whose operations are term operations of ℛ . Then a symmetric difference of ℬ together with the meet operation recover the original ring operations of ℛ . The aim of this paper is to show for what a ring ℛ a similar construction is possible. Of course, we do not construct a Boolean algebra but only so-called lattice-like structure which was introduced and treated by the authors in a previous paper. In particular,...

When is every order ideal a ring ideal?

Melvin Henriksen, Suzanne Larson, Frank A. Smith (1991)

Commentationes Mathematicae Universitatis Carolinae

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A lattice-ordered ring ℝ is called an if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those f -rings ℝ such that ℝ / 𝕀 is contained in an f -ring with an identity element that is a strong order unit for some nil l -ideal 𝕀 of ℝ . In particular, if P ( ℝ ) denotes the set of nilpotent elements of the f -ring ℝ , then ℝ is an OIRI-ring if and only if ℝ / P ( ℝ ) is contained in an f -ring with an identity element that is a strong order unit. ...